#355832

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1540 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)(5,6,8,7)(10,13)(11,15)(17,18,21,25,22,26,29,32)(19,23,20,24,27,30,28,31);(9,10)(11,14)(12,13)(15,16)(17,19,21,20,22,27,29,28)(18,23,25,24,26,30,32,31);(2,4)(10,13)(14,16);(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,27,22,28,29,19)(18,24,25,30,26,31,32,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,g^2=i,h^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, 263028);