#10020534

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

4033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,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, 9927730);