#8772786

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

4266 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8679982);