#357231

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 264427);