#9175036

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

4290 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9082232);