#9508653

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=hi,e^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, 9415849);