#6802446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709642);