#10166163

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

4056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073359);