#6456904

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,27,29,24)(20,25,28,31)(22,26,30,32);(1,3,5,4)(2,6,8,7)(18,23)(19,28)(20,29)(22,30)(24,31)(25,27);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,24)(20,22)(21,29)(23,27)(25,32)(26,31)(28,30);(1,4,5,3)(2,6,8,7)(17,20,21,28)(18,25,23,31)(19,22,29,30)(24,32,27,26);(9,11,12,15)(10,14,13,16)(18,23)(24,27)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6364100);