#6438710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3171 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345906);