#8157664

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

4157 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(20,28)(23,30)(26,32)(29,31)(33,34,39,42)(35,41,38,40)(36,37,46,48)(43,47,45,44);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,39,38)(34,40,42,41)(36,43,46,45)(37,44,48,47);(2,7)(3,8)(9,13)(11,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,39,48)(34,36,42,46)(35,44,38,47)(40,43,41,45);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38,39,35)(34,41,42,40)(36,45,46,43)(37,47,48,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064860);