#8757384

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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)(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(26,31)(28,39)(29,41)(32,35)(36,49)(37,51)(46,53)(54,55);(1,2,4,6)(3,5,8,7)(13,21)(17,20)(19,22)(23,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,42)(40,46)(41,43)(44,49)(45,51)(50,55)(52,53)(54,56);(1,3,4,8)(2,5,6,7)(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28,30,39)(26,33,31,42)(27,36,38,49)(29,40,41,52)(32,44,35,47)(34,46,43,53)(37,50,51,56)(45,54,48,55);(9,14,15,11)(10,18,12,16)(13,22,21,19)(17,24,20,23)(25,29,30,41)(26,34,31,43)(27,37,38,51)(28,40,39,52)(32,45,35,48)(33,46,42,53)(36,50,49,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664580);