#372251

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

962 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=ef,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279447);