#6438709

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

3247 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)(9,10,14,18)(11,15,21,30)(12,16,24,33)(13,17,27,36)(19,28,37,39)(20,29,22,31)(23,32,38,40)(25,34,26,35);(5,6,8,7)(9,11,14,21)(10,15,18,30)(12,23,24,38)(13,20,27,22)(16,32,33,40)(17,29,36,31)(19,26,37,25)(28,35,39,34);(2,4)(5,7,8,6)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,35)(18,33)(20,23)(21,37)(22,38)(25,27)(29,32)(30,39)(31,40)(34,36);(11,20)(13,27)(15,29)(17,36)(19,23)(21,22)(25,26)(28,32)(30,31)(34,35)(37,38)(39,40);(11,21)(12,24)(15,30)(16,33)(20,22)(25,26)(29,31)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,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.

Relationships

maximal subgroup of	#72316094, #72316120, #72316135
maximal normal subgroup of	#72316094, #72316120, #72316135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345905);