#6516844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500019
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3484 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:

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

Polycyclic presentation:

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

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, 6424040);