#386765

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efg,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293961);