#6375193

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1492 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,23,21,26,19,24)(20,28,30,32,29,25,27,31);(9,10,13,14)(11,16,12,15)(17,19,21,22)(18,23,26,24)(20,27,29,30)(25,31,28,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,21,22)(18,24,26,23)(20,27,29,30)(25,32,28,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,32)(26,28);(20,29)(25,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,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, 6282389);