#375578

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

894 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)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26,30,33,29,32,38,28)(27,31,36,34,35,39,40,37);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(18,21)(19,23)(25,27,29,35)(26,31,32,39)(28,37,33,34)(30,36,38,40);(10,13)(14,15)(17,19,20,23)(18,22,21,24)(25,28,38,32,29,33,30,26)(27,34,40,31,35,37,36,39);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(18,21)(22,24)(27,35)(31,39)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=gh,d^2=f,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, 282774);