#9814618

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

3961 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)(3,5)(4,9)(6,15)(7,8)(10,12)(11,14)(13,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34)(35,45)(36,39)(37,46)(38,42)(40,44)(41,48)(43,47);(1,3)(2,5)(4,12)(6,13)(7,11)(8,14)(9,10)(15,16)(19,25)(23,26)(27,29)(31,32)(33,35)(34,41)(36,46)(37,42)(38,43)(39,47)(40,48)(44,45);(2,8)(3,11)(4,13)(10,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36,40,38)(34,42,44,39)(35,46,48,43)(37,45,47,41);(1,4,7,13)(2,9,8,16)(3,12,11,6)(5,10,14,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,37,40,47)(34,43,44,46)(35,42,48,39)(36,41,38,45);(1,5)(2,3)(4,10)(6,16)(7,14)(8,11)(9,12)(13,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,38,40,36)(34,42,44,39)(35,43,48,46)(37,45,47,41);(1,6,7,12)(2,10,8,15)(3,13,11,4)(5,9,14,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,42)(34,38,44,36)(35,47,48,37)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=h,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, 9721814);