#8934180

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

4114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=gi,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, 8841376);