#6672763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(13,14,18,16)(15,23,27,20)(17,28,30,21)(19,22,32,29)(24,26,36,33)(25,34,41,37)(31,42,43,35)(38,44,39,40);(6,9)(7,11)(13,15,18,27)(14,20,16,23)(17,25,30,41)(19,33,32,26)(21,34,28,37)(22,36,29,24)(31,44,43,40)(35,39,42,38);(2,4)(13,16,18,14)(15,24,27,36)(17,21,30,28)(19,22,32,29)(20,33,23,26)(25,39,41,38)(31,42,43,35)(34,40,37,44);(6,9)(10,12)(13,17,18,30)(14,21,16,28)(15,25,27,41)(19,31,32,43)(20,34,23,37)(22,35,29,42)(24,38,36,39)(26,40,33,44);(5,7,8,11)(6,10,9,12)(15,26)(19,32)(20,24)(22,29)(23,36)(25,40)(27,33)(31,43)(34,38)(35,42)(37,39)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579959);