#6963379

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1220 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)(11,17)(14,18)(19,20)(22,23)(25,26,29,35,30,34,42,32)(27,31,37,46,38,45,52,43)(28,33,41,49,40,50,54,47)(36,44,51,56,39,48,53,55);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27,30,38)(26,31,34,45)(28,39,40,36)(29,37,42,52)(32,43,35,46)(33,48,50,44)(41,53,54,51)(47,56,49,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(25,28,30,40)(26,33,34,50)(27,36,38,39)(29,41,42,54)(31,44,45,48)(32,47,35,49)(37,51,52,53)(43,55,46,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(26,34)(28,40)(31,45)(32,35)(36,39)(41,54)(43,46)(51,53)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fh,d^2=fhi,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, 6870575);