#356830

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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

1354 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)(10,14)(15,16)(17,18,21,25,20,22,26,29)(19,23,28,30,27,24,31,32);(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(18,22)(23,24)(25,29)(30,32);(2,4)(9,11,13,12)(10,15,14,16)(17,19,26,31,20,27,21,28)(18,23,29,32,22,24,25,30);(9,12,13,11)(10,15,14,16)(17,19,26,31,20,27,21,28)(18,24,29,30,22,23,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264026);