#6381290

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1468 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=dfi,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, 6288486);