#369530

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#2366
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 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)(9,10,14,17,15,20,13,19)(11,21,24,32,25,31,22,16)(12,26,29,34,27,33,28,18)(23,37,38,40,39,30,36,35);(2,5)(6,7)(9,11,15,25)(10,16)(12,23,27,39)(13,22,14,24)(17,21)(18,30)(19,31)(20,32)(26,35)(28,36,29,38)(33,40)(34,37);(10,17,20,19)(11,22,25,24)(12,27)(18,33,34,26)(23,38,39,36)(28,29)(30,37)(35,40);(1,3,4,8)(2,6,5,7)(9,12,13,28,15,27,14,29)(10,18,17,26,20,34,19,33)(11,23,22,36,25,39,24,38)(16,30,21,35,32,37,31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fg,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 276726);