#10143766

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

4183 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35)(34,40)(36,38)(37,45)(39,46)(41,48)(42,47)(43,44);(2,6)(7,9)(8,12)(15,16)(18,25)(20,24)(26,30)(27,29)(33,36,39,37)(34,41,43,47)(35,38,46,45)(40,48,44,42);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,37,39,36)(34,41,43,47)(35,45,46,38)(40,48,44,42);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,34)(35,40)(36,41)(37,47)(38,48)(39,43)(42,45)(44,46);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,38,39,45)(34,42,43,48)(35,36,46,37)(40,47,44,41)

Polycyclic presentation:

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

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, 10050962);