#8934182

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

4134 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841378);