#391847

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
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

992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299043);