#6480831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=hi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 6388027);