#6597026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2144 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,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, 6504222);