#8772814

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

4065 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(19,22)(24,27)(29,31)(30,32)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(1,2,6,4)(3,7,11,8)(5,12,13,9)(10,15,16,14)(17,18,23,28)(19,27,22,24)(20,25,26,21)(29,30,31,32)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,2)(3,7,11,8)(5,12,13,9)(10,14,16,15)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 8680010);