#9685573

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

3950 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 9592769);