#6809822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717018);