#8115460

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 8022656);