#372250

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄²

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

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 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)(25,26,30,33,29,32,34,27)(28,36,38,35,37,31,39,40);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,27,34,32,29,33,30,26)(28,35,39,36,37,40,38,31);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16)(18,22)(19,24)(25,28,29,37)(26,31,32,36)(27,35,33,40)(30,38,34,39);(1,4,5,3)(2,7,8,6)(18,22)(21,23)(28,37)(31,36)(35,40)(38,39)

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^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279446);