#9936337

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

4116 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)(3,4)(5,8)(6,14)(7,9)(10,15)(11,16)(12,13)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34)(35,37)(36,43)(38,41)(39,46)(40,47)(42,48)(44,45);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(2,9)(4,8)(10,14)(12,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,43)(47,48);(1,5,7,3)(2,8,9,4)(6,11,15,13)(10,12,14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,34)(35,37)(36,39)(38,41)(40,47)(42,45)(43,46)(44,48);(1,6,7,15)(2,10,9,14)(3,11,5,13)(4,12,8,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9843533);