#8741961

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

4107 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)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,32,30)(33,34)(35,44)(36,41)(37,40)(38,42)(39,43)(45,46)(47,48);(2,7)(3,10)(8,14)(11,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,39,37)(34,40,43,44)(36,45,42,47)(38,46,41,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,22,29)(18,23,26,24)(20,27,25,32)(21,28,31,30)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,48,44,46);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,36)(34,41)(35,45)(37,47)(38,43)(39,42)(40,48)(44,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,21)(19,27)(22,25)(23,28)(24,30)(26,31)(29,32)(33,37,39,35)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,47,44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649157);