#8115496

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(26,31)(28,39)(29,41)(32,36)(37,47)(38,48)(46,53)(54,56);(1,2,4,6)(3,5,8,7)(11,17)(14,21)(18,23)(22,24)(26,31)(27,35)(28,39)(34,43)(38,48)(40,52)(44,49)(54,56);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,49)(38,50)(39,42)(40,46)(41,43)(44,47)(45,48)(51,56)(52,53)(54,55);(1,3,4,8)(2,5,6,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,30,35)(26,32,31,36)(28,37,39,47)(29,38,41,48)(33,44,42,49)(34,45,43,50)(40,51,52,55)(46,54,53,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,28,30,39)(26,33,31,42)(27,37,35,47)(29,40,41,52)(32,44,36,49)(34,46,43,53)(38,51,48,55)(45,54,50,56);(25,29,30,41)(26,34,31,43)(27,38,35,48)(28,40,39,52)(32,45,36,50)(33,46,42,53)(37,51,47,55)(44,54,49,56)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022692);