#9850762

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

3959 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9757958);