#9160824

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
prime radical	#57605	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(18,22)(19,25)(24,27)(26,28)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(2,6)(7,9)(8,12)(15,16)(17,19,21,25)(18,23,22,30)(20,26,29,28)(24,31,27,32)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068020);