#6378914

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

1650 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,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, 6286110);