#6407753

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(3,7)(4,5)(6,8)(9,10,13,15,12,14,11,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(2,5)(6,7)(10,14)(15,16)(19,26)(22,25)(27,32)(29,31);(1,3,4,8)(2,6,5,7)(9,11,12,13)(10,15,14,16)(18,23)(19,26)(24,30)(27,32);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=df

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, 6314949);