#8755186

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

4135 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:

(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,6)(4,12)(7,9)(11,16)(18,23)(21,24)(25,28)(29,31)(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)(19,26)(25,28)(27,32)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(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=ghi,e^2=h,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, 8662382);