#316304

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=h,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223500);