#6948374

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fi,d^2=fh,e^2=fi,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, 6855570);