#8691012

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

4206 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598208);