#381413

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=de

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, 288609);