#6490977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3038 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398173);