#8741942

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

3943 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)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,32,30)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(2,6)(7,9)(8,12)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(2,6)(7,9)(8,12)(15,16)(17,19,22,29)(18,23,26,24)(20,27,25,32)(21,28,31,30)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,21)(19,27)(22,25)(23,28)(24,30)(26,31)(29,32);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649138);