#6864819

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=hi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6772015);