#362953

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di

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, 270149);