#9863439

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

3935 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)(3,8)(4,5)(6,15)(7,11)(9,14)(10,12)(13,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(34,40)(36,45)(41,46)(43,44);(1,3,7,13)(2,8,11,16)(4,12,14,6)(5,10,9,15)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34)(35,41)(36,42)(37,46)(38,43)(39,40)(44,48)(45,47);(1,4,7,14)(2,9,11,5)(3,6,13,12)(8,10,16,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,37)(34,41,40,46)(36,44,45,43)(38,42,48,47);(1,4)(2,5)(3,12)(6,13)(7,14)(8,10)(9,11)(15,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,42)(35,44)(37,43)(38,46)(39,45)(40,47)(41,48);(1,5)(2,4)(3,10)(6,16)(7,9)(8,12)(11,14)(13,15)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,35)(34,41,40,46)(36,43,45,44)(38,42,48,47);(1,6,7,12)(2,10,11,15)(3,4,13,14)(5,8,9,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,48)(34,43,40,44)(35,42,37,47)(36,46,45,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=hi,e^2=ghi,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, 9770635);