#6467831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	#28584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(18,33)(19,24)(20,28)(23,30)(26,35)(27,39)(29,40)(31,37)(32,38)(34,43)(36,41)(42,44);(2,3)(13,15,18,28,19,29,30,41)(14,20,23,25,24,36,33,40)(16,26,32,43,31,42,17,27)(21,34,22,35,37,39,38,44);(1,2,4,3)(6,9)(10,12)(13,16,19,31)(14,21,24,37)(15,26,29,42)(17,18,32,30)(20,34,36,39)(22,33,38,23)(25,35,40,44)(27,28,43,41);(1,3,4,2)(13,17,19,32)(14,22,24,38)(15,27,29,43)(16,30,31,18)(20,35,36,44)(21,23,37,33)(25,39,40,34)(26,41,42,28);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,37)(22,38)(26,42)(27,43)(34,39)(35,44)

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=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375027);