#8934189

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

4111 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)(3,6)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(34,39)(36,46)(40,41)(43,44);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,34)(35,40)(36,47)(37,43)(38,39)(41,45)(42,46)(44,48);(2,3)(5,13)(7,9)(11,16)(19,27)(20,30)(23,31)(24,29)(33,35,38,45)(34,40,39,41)(36,44,46,43)(37,47,48,42);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,41)(36,43)(37,42)(38,45)(39,40)(44,46)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,36)(34,42)(35,44)(37,40)(38,46)(39,47)(41,48)(43,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,37,38,48)(34,43,39,44)(35,42,45,47)(36,40,46,41)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=h,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, 8841385);