#9851257

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
prime radical	#49899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=ghi,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, 9758453);