#9851270

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	#41087	C₂
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:

(1,2,7,10)(3,8,14,5)(4,6,16,9)(11,15,12,13)(17,18)(19,29)(20,23)(21,30)(22,26)(24,28)(25,32)(27,31)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,3,7,14)(2,8,10,5)(4,12,16,11)(6,13,9,15)(17,19)(18,25)(20,30)(21,26)(22,27)(23,31)(24,32)(28,29)(33,34,39,44)(35,40,47,48)(36,41,45,46)(37,38,43,42);(1,4,7,16)(2,9,10,6)(3,11,14,12)(5,15,8,13)(17,20,24,22)(18,26,28,23)(19,30,32,27)(21,29,31,25)(33,35,39,47)(34,40,44,48)(36,37,45,43)(38,46,42,41);(1,4)(2,6)(3,12)(5,15)(7,16)(8,13)(9,10)(11,14)(17,21,24,31)(18,27,28,30)(19,26,32,23)(20,25,22,29)(33,36,39,45)(34,41,44,46)(35,37,47,43)(38,48,42,40);(1,5)(2,3)(4,15)(6,12)(7,8)(9,11)(10,14)(13,16)(17,22,24,20)(18,26,28,23)(19,27,32,30)(21,29,31,25)(33,37)(34,42)(35,45)(36,47)(38,44)(39,43)(40,41)(46,48);(1,6)(2,4)(3,13)(5,11)(7,9)(8,12)(10,16)(14,15)(17,23,24,26)(18,22,28,20)(19,31,32,21)(25,27,29,30)(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=gh,c^2=gi,d^2=g,e^2=hi,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, 9758466);