#8127249

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,f^2=g,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^c=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, 8034445);