#6437755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	#34632	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4691 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)(9,10,13,18,14,17,26,31)(11,19,21,28,22,32,25,15)(12,16,24,30,23,29,36,39)(20,33,35,38,34,40,37,27);(5,6)(7,8)(9,11,14,22)(10,15,17,28)(12,20,23,34)(13,25,26,21)(16,27,29,38)(18,32,31,19)(24,37,36,35)(30,40,39,33);(2,4)(6,8)(9,12,14,23)(10,16,17,29)(11,20,22,34)(13,24,26,36)(15,27,28,38)(18,30,31,39)(19,33,32,40)(21,35,25,37);(6,8)(12,23)(16,29)(20,34)(24,36)(27,38)(30,39)(33,40)(35,37);(10,17)(15,28)(16,29)(18,31)(19,32)(27,38)(30,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316094, #72316207
maximal normal subgroup of	#72316094, #72316207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344951);