#6438711

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×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3219 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)(9,10,15,14)(11,16,21,27)(12,17,24,28)(13,18,26,29)(19,30,34,22)(20,31,35,25)(23,32,37,38)(33,39,40,36);(5,6,8,7)(9,11)(10,16)(12,22)(13,25)(14,27)(15,21)(17,19)(18,20)(23,33)(24,30)(26,31)(28,34)(29,35)(32,39)(36,38)(37,40);(2,4)(5,7,8,6)(9,12)(10,17)(11,19)(13,23)(14,28)(15,24)(16,30)(18,32)(20,33)(21,34)(22,27)(25,36)(26,37)(29,38)(31,39)(35,40);(9,13,15,26)(10,18,14,29)(11,20,21,35)(12,23,24,37)(16,31,27,25)(17,32,28,38)(19,33,34,40)(22,36,30,39);(11,21)(12,24)(16,27)(17,28)(20,35)(23,37)(25,31)(32,38)

Polycyclic presentation:

2^9|a^2=i,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, 6345907);