#9865950

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

3950 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34,39,36)(35,37,40,42)(38,44,47,41)(43,46,48,45);(2,7)(4,8)(9,15)(12,13)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(18,23)(22,29)(24,27)(30,31)(34,36)(35,40)(41,44)(43,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=i,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, 9773146);