#9723161

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

4118 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)(19,29)(20,23)(21,26)(22,27)(24,28)(25,31)(30,32)(33,34,40,37)(35,44,46,41)(36,47,38,42)(39,48,45,43);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(17,19,24,31)(18,25,28,29)(20,30,21,22)(23,27,26,32)(33,35)(34,41)(36,45)(37,44)(38,39)(40,46)(42,43)(47,48);(2,9)(3,14)(4,11)(12,16)(17,20,24,21)(18,23,28,26)(19,30,31,22)(25,27,29,32)(33,36,40,38)(34,42,37,47)(35,45,46,39)(41,43,44,48);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,21,24,20)(18,26,28,23)(19,22,31,30)(25,32,29,27)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,48,45,43);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,22)(18,27)(19,20)(21,31)(23,29)(24,30)(25,26)(28,32)(33,38,40,36)(34,42,37,47)(35,45,46,39)(41,48,44,43);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,23,24,26)(18,20,28,21)(19,27,31,32)(22,25,30,29)(33,39,40,45)(34,43,37,48)(35,38,46,36)(41,42,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,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, 9630357);