#382091

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 289287);