#6564692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#3301	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471888);