#380996

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
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

1186 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:

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

Polycyclic presentation:

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

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, 288192);