#384491

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 291687);