#6808920

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=hi,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, 6716116);