#6437674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4665 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)(5,6)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,28)(20,29)(21,30)(22,31)(23,33)(24,32)(25,35)(26,36)(27,34)(37,39)(38,40);(2,4)(9,11,14,21)(10,15,17,30)(12,22,24,37)(13,25,27,20)(16,31,32,39)(18,35,34,29)(19,26,38,23)(28,36,40,33);(6,8)(9,12,14,24)(10,16,17,32)(11,19,21,38)(13,26,27,23)(15,28,30,40)(18,36,34,33)(20,37,25,22)(29,39,35,31);(11,20)(13,27)(15,29)(18,34)(19,37)(21,25)(22,38)(23,26)(28,39)(30,35)(31,40)(33,36);(10,17)(15,30)(16,32)(18,34)(28,40)(29,35)(31,39)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315569, #72315572, #72315588
maximal normal subgroup of	#72315569, #72315572, #72315588
maximal quotient of	#72315759

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344870);