#6487532

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

4008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,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, 6394728);