#6383732

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

1414 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)(17,18,23,19,22,25,32,27)(20,28,29,30,21,24,31,26);(1,2)(3,7)(4,8)(5,6)(17,19,23,25,22,27,32,18)(20,26,29,28,21,30,31,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,21)(18,24,25,28)(19,26,27,30)(23,29,32,31);(1,3,5,4)(2,8,6,7)(17,21,22,20)(18,24,25,28)(19,26,27,30)(23,31,32,29);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(20,21)(24,28)(26,30)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^b=df,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, 6290928);