#6809818

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,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, 6717014);