#316316

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

713 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)(11,14)(15,16)(17,18)(19,23)(20,30)(21,26)(22,25)(24,27)(28,32)(29,31);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19,20,27,22,29,21,28)(18,23,30,24,25,31,26,32);(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,24,25,32)(20,28,21,27)(23,26,31,30);(2,6)(5,7)(9,10,12,13)(11,14,15,16)(18,25)(23,31)(24,32)(26,30)

Polycyclic presentation:

2^9|b^2=egi,c^2=h,d^2=hi,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, 223512);