#9508674

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

3978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=ghi,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, 9415870);