#6407481

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(2,6)(5,8)(10,13)(14,16)(18,23)(22,28)(24,30)(29,32);(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfg,d^b=df,e^b=ei,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, 6314677);