#6948341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1230 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,29,34,30,33,42,47)(27,31,37,45,36,44,52,55)(28,38,39,46,40,48,41,32)(35,49,51,54,50,56,53,43);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,36)(26,31,33,44)(28,35,40,50)(29,37,42,52)(32,43,46,54)(34,45,47,55)(38,49,48,56)(39,51,41,53);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28,30,40)(26,32,33,46)(27,35,36,50)(29,41,42,39)(31,43,44,54)(34,48,47,38)(37,53,52,51)(45,56,55,49);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(27,36)(31,44)(35,50)(37,52)(43,54)(45,55)(49,56)(51,53);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(26,33)(31,44)(32,46)(34,47)(38,48)(43,54)(45,55)(49,56)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6855537);