#355364

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,20,21,24,29,28)(19,23,27,25,26,30,32,31);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(2,4)(11,16)(13,15)(17,20,22,24,21,28,29,18)(19,25,27,30,26,31,32,23);(10,14)(13,15)(19,26)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262560);