#373303

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1224 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280499);