#6383589

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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	#34909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,24,26,23,27,22,32)(19,28,30,31,29,20,25,21);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,30,23,29,22,25)(18,20,26,21,27,28,32,31);(10,14)(13,15)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,30)(29,32);(2,5)(6,8)(11,16)(13,15)(17,21)(18,19)(20,22)(23,31)(24,28)(25,32)(26,30)(27,29);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,26,27,32)(19,25,29,30)(20,21,28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=fg,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290785);