#6467838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 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)(15,25)(16,21)(17,22)(18,34)(19,24)(20,28)(23,30)(26,39)(27,35)(29,41)(31,38)(32,40)(33,44)(36,42)(37,43);(2,3)(13,15,18,28,19,29,30,43)(14,20,23,25,24,37,34,41)(16,26,32,27,31,42,17,33)(21,35,22,39,38,44,40,36);(1,2,4,3)(6,9)(10,12)(13,16,19,31)(14,21,24,38)(15,26,29,42)(17,18,32,30)(20,35,37,44)(22,34,40,23)(25,39,41,36)(27,43,33,28);(1,3,4,2)(13,17,19,32)(14,22,24,40)(15,27,29,33)(16,30,31,18)(20,36,37,39)(21,23,38,34)(25,35,41,44)(26,28,42,43);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,38)(22,40)(26,42)(27,33)(35,44)(36,39)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375034);