#6575607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482803);