#9782781

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

4139 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,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, 9689977);