#9794293

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	#52637	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,24,21)(19,22,25,20)(23,29,31,26)(27,28,32,30)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,14)(15,16)(17,19)(18,22)(20,21)(23,27)(24,25)(26,30)(28,29)(31,32)(33,34,39,44)(35,40,47,48)(36,41,45,46)(37,38,43,42);(2,7)(4,9)(8,15)(12,16)(17,20,24,22)(18,25,21,19)(23,28,31,30)(26,32,29,27)(33,35,39,47)(34,40,44,48)(36,37,45,43)(38,46,42,41);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,24,18)(19,20,25,22)(23,29,31,26)(27,28,32,30)(33,36,39,45)(34,41,44,46)(35,37,47,43)(38,48,42,40);(1,5,6,14)(2,8,7,15)(3,10,11,13)(4,12,9,16)(17,22,24,20)(18,25,21,19)(23,28,31,30)(26,27,29,32)(33,37)(34,42)(35,45)(36,47)(38,44)(39,43)(40,41)(46,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,16)(10,14)(12,15)(17,23,24,31)(18,26,21,29)(19,27,25,32)(20,28,22,30)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,48,45,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=gh,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, 9701489);