#8772872

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
prime radical	#55034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,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, 8680068);