#6802471

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709667);