#8943979

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

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, 8851175);