#8757311

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^2=gi,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, 8664507);