#6576345

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6483541);