#6669898

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(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);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,34)(22,27,30,32)(24,33,37,46)(26,35,39,48)(28,40,31,42)(29,41,36,45)(38,47,49,52)(43,50,44,51);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(26,39)(27,40)(29,43)(30,31)(32,42)(35,48)(36,44)(38,49)(41,50)(45,51)(47,52);(6,12)(10,13)(11,18)(17,19)(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);(2,4)(7,15)(10,13)(14,20)(17,19)(24,37)(29,36)(33,46)(38,49)(41,45)(43,44)(47,52)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6577094);