#389636

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	#2361
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

723 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)(3,7)(4,5)(6,8)(9,10,14,20,16,13,15,19)(11,21,23,17,24,33,30,29)(12,18,27,36,28,26,31,35)(22,34,37,32,38,40,25,39);(9,11,14,23,16,24,15,30)(10,17,19,21,13,29,20,33)(12,25,27,22,28,37,31,38)(18,34,35,39,26,40,36,32);(2,5)(6,7)(9,12,14,27,16,28,15,31)(10,18,20,36,13,26,19,35)(11,22,23,37,24,38,30,25)(17,32,33,40,29,39,21,34);(1,3,4,8)(2,6,5,7)(9,13)(10,16)(11,21)(12,26)(14,19)(15,20)(17,23)(18,28)(22,34)(24,33)(25,39)(27,35)(29,30)(31,36)(32,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296832);