#8755194

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

4153 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662390);