#6948381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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)(25,26,32,36,31,37,30,38)(27,39,41,50,42,33,43,53)(28,35,44,34,29,46,45,51)(40,49,56,54,47,52,55,48);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,42)(26,33,37,39)(28,40,29,47)(30,43,32,41)(34,48,51,54)(35,52,46,49)(36,53,38,50)(44,56,45,55);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28)(26,34)(27,40)(29,31)(30,44)(32,45)(33,48)(35,36)(37,51)(38,46)(39,54)(41,55)(42,47)(43,56)(49,50)(52,53);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,31,28)(26,35,37,46)(27,40,42,47)(30,44,32,45)(33,49,39,52)(34,38,51,36)(41,56,43,55)(48,53,54,50);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(25,30,31,32)(26,36,37,38)(27,41,42,43)(28,44,29,45)(33,50,39,53)(34,35,51,46)(40,55,47,56)(48,49,54,52)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,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,e^d=ei

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, 6855577);