#508108

order	512
nilpotency class	4
isoclinism	isoclinic class of #508104
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29636
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#13360	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

797 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,13,18,41)(4,19,23,57)(5,10,25,40)(6,26,29,11)(8,27,33,16)(9,34,38,22)(14,46,50,61)(15,44,17,45)(20,37,58,42)(21,36,28,56)(24,59,60,39)(30,47,35,64)(31,54,32,52)(43,51,62,49)(48,55,53,63);(1,3,7,18)(2,8,12,33)(4,14,23,50)(5,15,25,17)(6,27,29,16)(9,35,38,30)(10,31,40,32)(11,41,26,13)(19,55,57,63)(20,47,58,64)(21,53,28,48)(22,49,34,51)(24,54,60,52)(36,46,56,61)(37,62,42,43)(39,45,59,44);(1,4)(2,9)(3,14)(5,22)(6,28)(7,23)(8,30)(10,19)(11,42)(12,38)(13,43)(15,51)(16,53)(17,49)(18,50)(20,24)(21,29)(25,34)(26,37)(27,48)(31,55)(32,63)(33,35)(36,39)(40,57)(41,62)(44,61)(45,46)(47,52)(54,64)(56,59)(58,60);(1,5,7,25)(2,10,12,40)(3,15,18,17)(4,20)(6,24,29,60)(8,31,33,32)(9,36)(11,39,26,59)(13,44,41,45)(14,47)(16,52,27,54)(19,37)(21,22)(23,58)(28,34)(30,61)(35,46)(38,56)(42,57)(43,63)(48,51)(49,53)(50,64)(55,62)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415304);