#384364

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

995 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)(4,6)(5,8)(9,10,15,20,16,21,14,37)(11,22,25,17,27,29,26,38)(12,18,31,39,32,40,30,24)(13,33,35,23,36,28,34,19);(9,11)(10,17)(12,28)(13,18)(14,26)(15,25)(16,27)(19,31)(20,29)(21,38)(22,37)(23,30)(24,34)(32,33)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,23)(13,29)(14,30)(15,31)(16,32)(17,34)(19,27)(20,39)(21,40)(22,36)(24,37)(25,28)(26,33)(35,38);(2,6)(5,7)(9,13,14,34,16,36,15,35)(10,19,20,33,21,23,37,28)(11,24,25,18,27,39,26,40)(12,29,30,17,32,22,31,38)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=e,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,e^b=ei,g^b=gi,g^c=gi

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, 291560);