#8750857

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

4028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658053);