#387608

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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

818 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)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,31,29,33,27,34)(28,36,38,40,37,32,35,39);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,22)(19,23)(25,28,29,37)(26,32,33,36)(27,35,30,38)(31,39,34,40);(1,5,6,4)(2,7,3,8)(10,13)(14,16)(18,22)(21,24)(26,33)(28,37)(31,34)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 294804);