#6383129

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei

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, 6290325);