#9545803

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

4141 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,7)(4,14)(8,11)(10,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,39,36)(34,40,38,41)(37,44,45,43)(42,48,46,47);(2,7)(3,12)(4,14)(5,16)(19,25)(20,28)(22,26)(23,30)(34,38)(35,36)(37,45)(47,48);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,36,39,35)(34,41,38,40)(37,43,45,44)(42,47,46,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32)(33,37,39,45)(34,42,38,46)(35,43,36,44)(40,47,41,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,20,21,28)(18,23,24,30)(19,27,25,31)(22,29,26,32)(33,38,39,34)(35,41,36,40)(37,46,45,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=i,e^2=gi,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,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452999);