#6385619

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	#2688	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=dfi,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, 6292815);