#379687

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=egh,f^2=i,h^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, 286883);