#500818

order	512
nilpotency class	4
isoclinism	isoclinic class of #500814
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24260
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3018	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

993 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408014);