#9815153

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

4315 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)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(4,13)(8,9)(10,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(18,19)(23,25)(24,26)(30,31)(34,36)(39,41)(40,44)(46,47);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,e^2=g,f^2=h,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, 9722349);