#9481377

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

4196 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,31,34)(27,35,38,28)(29,32,41,45)(30,43,44,33)(36,47,49,39)(37,40,51,48)(42,54,53,46)(50,52,56,55);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(29,41)(32,45)(36,49)(39,47)(42,53)(46,54)(50,56)(52,55);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(25,27,31,38)(26,28,34,35)(29,36,41,49)(30,37,44,51)(32,39,45,47)(33,40,43,48)(42,50,53,56)(46,52,54,55);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,27)(29,39)(30,40)(31,35)(32,36)(33,37)(34,38)(41,47)(42,52)(43,51)(44,48)(45,49)(46,50)(53,55)(54,56);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29)(26,32)(27,36)(28,39)(30,42)(31,41)(33,46)(34,45)(35,47)(37,50)(38,49)(40,52)(43,54)(44,53)(48,55)(51,56);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,44)(26,33,34,43)(27,37,38,51)(28,40,35,48)(29,42,41,53)(32,46,45,54)(36,50,49,56)(39,52,47,55)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388573);