#6422056

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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	#52096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6329252);