#503233

order	512
nilpotency class	4
isoclinism	isoclinic class of #503229
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25771
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#24287	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1085 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,18,40)(4,9,22,35)(5,23,14,10)(6,25,27,11)(8,28,31,16)(15,38,45,63)(17,47,26,39)(19,48,50,64)(20,51,41,33)(21,34,53,56)(24,36,42,54)(29,57,59,44)(30,46,37,55)(32,60,61,49)(43,62,52,58);(1,3,14,28,7,18,5,16)(2,8,23,40,12,31,10,13)(4,19,41,60,22,50,20,49)(6,26,42,46,27,17,24,55)(9,32,51,64,35,61,33,48)(11,37,54,39,25,30,36,47)(15,43,57,53,45,52,44,21)(29,58,63,56,59,62,38,34);(1,4)(2,9)(3,15)(5,20)(6,21,27,53)(7,22)(8,29)(10,33)(11,34,25,56)(12,35)(13,38)(14,41)(16,44)(17,19,26,50)(18,45)(23,51)(24,52,42,43)(28,57)(30,32,37,61)(31,59)(36,62,54,58)(39,48,47,64)(40,63)(46,49,55,60);(1,3,14,28,7,18,5,16)(2,8,23,40,12,31,10,13)(4,15,41,57,22,45,20,44)(6,17,42,55,27,26,24,46)(9,29,51,63,35,59,33,38)(11,30,54,47,25,37,36,39)(19,43,60,53,50,52,49,21)(32,58,64,56,61,62,48,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341810, #72341981, #72342030
maximal normal subgroup of	#72341810, #72341981, #72342030

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410429);