#9724825

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

4043 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18)(19,26)(20,22)(21,25)(23,27)(24,31)(28,32)(29,30)(33,34)(35,41)(36,40)(37,39)(38,42)(43,47)(44,46)(45,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(2,9)(3,13)(4,15)(11,16)(17,20,23,21)(18,22,27,25)(19,28,31,29)(24,30,26,32)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,19)(18,26)(20,28)(21,29)(22,32)(23,31)(24,27)(25,30)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,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, 9632021);