#9711900

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

4098 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9619096);