#9508673

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

4142 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(33,34,37,41)(35,38,43,39)(36,40,45,48)(42,46,44,47);(2,7)(4,15)(8,11)(10,13)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(2,7)(3,12)(4,15)(5,16)(18,23)(20,27)(21,30)(26,28)(33,35,37,43)(34,39,41,38)(36,44,45,42)(40,46,48,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36,37,45)(34,40,41,48)(35,42,43,44)(38,46,39,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(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=gh,d^2=hi,e^2=h,f^2=g,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, 9415869);