#9913281

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

4245 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35)(34,39)(36,37)(38,44)(40,47)(41,45)(42,48)(43,46);(2,7)(3,10)(8,15)(11,12)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,38,46)(34,40,42,45)(35,43,44,37)(39,41,48,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36)(34,40)(35,37)(38,46)(39,47)(41,48)(42,45)(43,44);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,41)(35,36)(38,43)(39,45)(40,48)(42,47)(44,46);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820477);