#256261

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

926 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(2,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18,22,26)(19,28,29,31)(20,25,21,24)(23,27,32,30);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(14,16)(17,19)(18,23)(20,30)(21,27)(22,29)(24,28)(25,31)(26,32);(2,5)(6,8)(19,29)(23,32)(27,30)(28,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163457);