#9782824

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
prime radical	#45467	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,27,23)(20,24,29,32)(26,30,31,28)(33,34,39,44)(35,45,46,40)(36,41,37,42)(38,43,48,47);(3,9)(6,13)(10,15)(14,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,35,39,46)(34,40,44,45)(36,43,37,47)(38,42,48,41);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,27)(18,23,25,21)(20,26,29,31)(24,28,32,30)(33,36,39,37)(34,41,44,42)(35,43,46,47)(38,45,48,40);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,24)(19,26)(21,30)(22,29)(23,28)(25,32)(27,31)(33,37,39,36)(34,42,44,41)(35,43,46,47)(38,45,48,40);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,19)(20,28)(22,23)(24,31)(25,27)(26,32)(29,30)(33,38)(34,43)(35,42)(36,45)(37,40)(39,48)(41,46)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^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, 9690020);