#6579295

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=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, 6486491);