#10143784

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	#57993	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(17,18,21,23)(19,25,27,22)(20,24,30,29)(26,32,28,31)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,6)(7,9)(8,12)(15,16)(18,23)(22,25)(24,29)(31,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,29)(21,23)(24,30)(25,27)(26,32)(28,31)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,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, 10050980);