#10247120

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

4160 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:

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

Polycyclic presentation:

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

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, 10154316);