#6579452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1995 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)(15,19,25,34)(16,21,29,35)(18,22,30,36)(23,33,26,31)(24,32,40,43)(27,38,28,37)(39,44,41,42);(1,2,4,3)(6,9)(10,12)(13,15,17,25)(14,19,20,34)(16,24,29,40)(18,23,30,26)(21,32,35,43)(22,33,36,31)(27,41,28,39)(37,44,38,42);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,20)(15,23)(16,27)(18,30)(19,31)(21,37)(24,40)(25,26)(28,29)(33,34)(35,38)(42,44);(13,16,17,29)(14,21,20,35)(15,24,25,40)(18,28,30,27)(19,32,34,43)(22,37,36,38)(23,39,26,41)(31,42,33,44);(15,23)(16,28)(18,30)(19,33)(21,37)(22,36)(25,26)(27,29)(31,34)(35,38)(39,41)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6486648);