#6516756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3370 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)(11,20,24,36)(12,17)(13,28)(14,18)(15,19)(16,23,31,22)(21,38,25,37)(26,33)(27,35)(29,34)(30,39,32,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,32)(18,36)(19,31)(20,28)(21,26)(27,39)(29,40)(30,33)(34,37)(35,38);(2,5)(3,7)(9,12)(10,17)(11,21)(13,27)(14,29)(15,26)(16,30)(18,34)(19,33)(20,37)(22,39)(23,40)(24,25)(28,35)(31,32)(36,38);(1,3,4,7)(2,6,5,8)(12,26)(17,33)(21,25)(27,29)(30,32)(34,35)(37,38)(39,40);(2,5)(6,8)(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,27,26,29)(16,20,31,36)(17,34,33,35)(21,39,25,40)(30,37,32,38)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,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, 6423952);