#10480810

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,8)(3,4)(6,10)(12,14)(20,28)(23,29)(26,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2,7,8)(3,11,4,9)(5,10,16,6)(12,13,14,15)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(34,39)(40,42)(41,46)(47,48);(1,3,7,4)(2,9,8,11)(5,12,16,14)(6,13,10,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,4,7,3)(2,9,8,11)(5,12,16,14)(6,15,10,13)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5)(2,10)(3,12)(4,14)(6,8)(7,16)(9,15)(11,13)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,6,7,10)(2,5,8,16)(3,13,4,15)(9,12,11,14)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,37,38,35)(34,42,39,40)(36,43,44,45)(41,47,46,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10388006);