#362813

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1458 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)(11,14)(12,13)(15,16)(17,18,21,26,22,25,20,24)(19,27,28,32,29,23,30,31);(1,3,4,8)(2,5,6,7)(10,13)(11,15);(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,6)(5,7)(10,13)(14,16)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

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

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, 270009);