#392296

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi

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, 299492);