#10023161

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

4217 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)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,39,36)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,47,46,42);(2,6)(7,10)(8,12)(14,16)(18,23)(19,26)(24,29)(27,32)(33,34,39,36)(35,40,45,43)(37,46,44,41)(38,42,48,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36,39,34)(35,40,45,43)(37,46,44,41)(38,47,48,42);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,41)(35,38)(36,46)(39,44)(40,47)(42,43)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,41)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,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, 9930357);