#9070541

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

4098 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)(3,9)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(33,34,40,44)(35,37,41,36)(38,42,43,39)(45,46,48,47);(2,6)(7,9)(8,12)(15,16)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,29,32)(33,35)(34,37)(36,44)(38,48)(39,46)(40,41)(42,47)(43,45);(2,6)(7,9)(8,12)(15,16)(18,23)(20,27)(25,31)(28,32)(33,36,40,37)(34,41,44,35)(38,46,43,47)(39,48,42,45);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,19)(18,20)(21,29)(22,24)(23,27)(25,32)(26,30)(28,31)(33,37,40,36)(34,41,44,35)(38,47,43,46)(39,48,42,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,29)(26,31)(30,32)(33,38,40,43)(34,42,44,39)(35,45,41,48)(36,47,37,46);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,29)(27,32)(33,39,40,42)(34,43,44,38)(35,46,41,47)(36,48,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8977737);