#6378969

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

1358 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)(17,18,21,25,22,20,24,29)(19,23,27,30,26,28,31,32);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,28)(21,27)(22,26)(24,31)(25,30)(29,32);(3,8)(5,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,26)(24,31)(25,30)(29,32);(2,6)(5,7)(19,26)(23,28)(27,31)(30,32);(9,12,13,11)(10,14,15,16)(17,20,21,29,22,18,24,25)(19,23,27,30,26,28,31,32)

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=cf,d^b=di,e^b=ei,d^c=di,e^c=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, 6286165);