#6444263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351459);