#363797

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

1110 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)(11,15)(12,14)(13,16)(19,25)(20,28)(22,29)(24,30)(26,31)(27,32);(1,2,4,3)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,24)(21,23)(22,27)(25,31)(28,30)(29,32);(11,16)(13,15)(17,19,20,25,21,27,28,32)(18,22,24,29,23,26,30,31);(1,3,4,2)(10,14)(13,15)(18,23)(22,26)(24,30)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=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, 270993);