#6378957

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	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^b=ei,d^c=di,e^c=ei,e^d=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, 6286153);