#6467529

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=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, 6374725);