#409604

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

892 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=f,d^2=ei,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

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, 316800);