#8114397

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
prime radical	#52773	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021593);