#6674442

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=f,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, 6581638);