#9232799

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 9139995);