#9210295

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=i,e^2=gh,f^2=ghi,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, 9117491);