#9015400

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

4241 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34)(35,40)(36,39)(37,41)(38,42)(43,45)(44,47)(46,48);(2,6)(7,9)(8,12)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,36)(37,43)(38,39)(40,42)(41,44)(45,48)(46,47);(2,6)(3,10)(4,13)(14,15)(18,20)(19,25)(21,29)(30,32)(33,36,38,40)(34,39,42,35)(37,45,46,44)(41,43,48,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,46)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,35)(34,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)

Polycyclic presentation:

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

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, 8922596);