#10023176

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

4270 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9930372);