#6443911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351107);