#381282

order	512
nilpotency class	3
isoclinism	isoclinic class of #195981
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

956 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)(17,18,23,27,22,26,21,25)(19,24,28,31,20,29,30,32);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,19)(18,24)(20,22)(21,30)(23,28)(25,32)(26,29)(27,31);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,24,26,29)(21,30,23,28)(25,31,27,32);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(17,21,22,23)(18,25,26,27)(19,28,20,30)(24,31,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288478);