#9668372

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

4218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575568);