#6800384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1192 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,30,35)(27,37,40,32)(28,41,43,50)(29,36,31,34)(33,45,48,42)(38,52,54,56)(39,47,44,49)(46,55,51,53);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,38)(29,44)(30,40)(31,39)(33,46)(34,49)(35,37)(36,47)(41,56)(42,55)(43,54)(45,53)(48,51)(50,52);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,30,43)(26,33,35,48)(27,38,40,54)(29,42,31,45)(32,46,37,51)(34,41,36,50)(39,53,44,55)(47,52,49,56);(10,17)(15,19)(16,18)(23,24)(25,29,30,31)(26,34,35,36)(27,39,40,44)(28,42,43,45)(32,47,37,49)(33,41,48,50)(38,53,54,55)(46,52,51,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,40)(32,37)(38,54)(39,44)(46,51)(47,49)(52,56)(53,55)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6707580);