#6547016

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(21,22,26,32,27,30,25,31)(23,28,36,43,34,44,35,42)(24,37,38,45,39,46,40,29)(33,47,49,51,48,52,50,41);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,34)(22,28,30,44)(24,33,39,48)(25,35,26,36)(29,41,45,51)(31,42,32,43)(37,47,46,52)(38,49,40,50);(6,10)(11,17)(12,18)(19,20)(21,24,27,39)(22,29,30,45)(23,33,34,48)(25,38,26,40)(28,41,44,51)(31,37,32,46)(35,49,36,50)(42,47,43,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,30)(23,34)(29,45)(31,32)(33,48)(35,36)(37,46)(49,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,31,30,32)(23,35,34,36)(24,38,39,40)(28,42,44,43)(29,37,45,46)(33,49,48,50)(41,47,51,52)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6454212);