#409574

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

854 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26,31,37)(27,38,40,34)(28,32,39,33)(29,36,30,35);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,28,31,40,30,39)(26,32,35,38,37,33,36,34);(9,12,13,11)(10,14,15,16)(19,23)(21,24)(25,27,29,28,31,40,30,39)(26,33,35,34,37,32,36,38);(1,4,5,3)(2,6,7,8)(25,28,29,40,31,39,30,27)(26,34,35,32,37,38,36,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=efh,c^2=fi,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316770);