#6675737

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,31)(23,32,27,28)(24,35,37,46)(25,38,39,30)(29,41,44,33)(34,42,40,45)(36,49,50,48)(43,47,51,52);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,27)(22,28,31,32)(24,33,37,41)(25,34,39,40)(29,35,44,46)(30,42,38,45)(36,47,50,52)(43,48,51,49);(2,4)(21,24,26,37)(22,29,31,44)(23,33,27,41)(25,36,39,50)(28,35,32,46)(30,43,38,51)(34,47,40,52)(42,48,45,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,39)(22,30,31,38)(23,34,27,40)(24,36,37,50)(28,42,32,45)(29,43,44,51)(33,47,41,52)(35,48,46,49);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(25,39)(30,38)(34,40)(36,50)(42,45)(43,51)(47,52)(48,49)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fi,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, 6582933);