#9726169

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

4091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633365);