#9232800

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

4153 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9139996);