#9685526

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

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9592722);