#6401802

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,22,23,28,21,27)(19,29,20,26,30,25,31,32);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,23,30)(18,25,28,29)(20,21,31,24)(22,32,27,26);(2,6)(5,8)(10,13)(14,16)(17,20)(18,26)(19,21)(22,25)(23,31)(24,30)(27,29)(28,32);(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,28,22)(19,20,30,31)(25,32,29,26);(10,13)(14,16)(17,22,24,28,23,27,21,18)(19,26,20,25,30,32,31,29)

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=ef,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308998);