#6802894

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	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

1242 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)(25,26,30,36,31,35,29,34)(27,32,38,47,39,48,44,51)(28,40,42,50,43,52,45,33)(37,53,55,49,41,56,54,46);(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)(28,41)(29,44)(30,38)(31,39)(33,49)(34,51)(35,48)(36,47)(37,43)(40,56)(42,54)(45,55)(46,50)(52,53);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,42)(30,45)(31,43)(32,46)(34,40)(35,50)(36,52)(38,54)(39,41)(44,55)(47,56)(48,49)(51,53);(10,17)(15,19)(16,18)(23,24)(25,29,31,30)(26,34,35,36)(27,38,39,44)(28,42,43,45)(32,47,48,51)(33,40,50,52)(37,54,41,55)(46,56,49,53);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(26,35)(27,39)(33,50)(34,36)(37,41)(38,44)(40,52)(54,55)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=i,e^2=fhi,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, 6710090);