#6459602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 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)(3,5)(4,6)(7,8)(9,10,15,19,16,20,14,18)(11,21,23,31,25,32,24,13)(12,26,29,17,30,33,28,34)(22,36,37,40,39,27,38,35);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,22)(14,23)(15,24)(16,25)(17,27)(18,21)(19,32)(20,31)(26,35)(28,37)(29,38)(30,39)(33,40)(34,36);(2,6)(3,7)(9,12,15,29,16,30,14,28)(10,17,18,26,20,34,19,33)(11,22,24,38,25,39,23,37)(13,27,21,35,31,36,32,40);(3,7)(5,8)(9,13)(10,11)(12,27)(14,32)(15,21)(16,31)(17,22)(18,24)(19,23)(20,25)(26,38)(28,40)(29,35)(30,36)(33,37)(34,39);(9,14,16,15)(10,18,20,19)(11,23,25,24)(12,28,30,29)(13,21,31,32)(17,26,34,33)(22,37,39,38)(27,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366798);