#6407905

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#2819	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(10,13)(11,15)(17,18,23,28,22,27,30,31)(19,29,32,25,26,24,20,21);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,28)(22,26)(23,32)(25,31)(27,29);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,27,21)(19,30,26,23)(24,31,29,28);(2,6)(5,8)(10,13)(14,16)(19,26)(20,32)(21,25)(24,29);(17,21,23,29,22,25,30,24)(18,26,28,20,27,19,31,32)

Polycyclic presentation:

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

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, 6315101);