#8772645

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

4073 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(17,18)(19,27)(20,30)(21,31)(22,26)(23,29)(24,28)(25,32)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(2,7)(5,8)(9,15)(13,16)(17,19)(18,23)(20,21)(22,29)(24,31)(25,30)(26,27)(28,32)(34,39)(36,45)(40,41)(42,48);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,18,22,26)(19,23,29,27)(20,30,28,24)(21,25,32,31)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,20,22,28)(18,24,26,30)(19,21,29,32)(23,31,27,25)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,4,6,14)(2,9,7,15)(3,10,11,12)(5,13,8,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,30)(31,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,21)(18,25)(19,28)(20,29)(22,32)(23,24)(26,31)(27,30)(33,37)(34,36)(35,43)(38,46)(39,45)(40,42)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,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,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, 8679841);