#6774411

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1191 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)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,34)(27,36,38,32)(28,40,42,50)(29,35,31,44)(33,45,49,41)(37,52,53,56)(39,48,43,47)(46,55,51,54);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,30,38)(26,32,34,36)(28,37,42,53)(29,43,31,39)(33,46,49,51)(35,48,44,47)(40,56,50,52)(41,55,45,54);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,46)(34,49)(35,50)(36,51)(38,53)(39,54)(40,44)(43,55)(47,56)(48,52);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(27,38)(33,49)(35,44)(37,53)(39,43)(40,50)(54,55);(2,5)(6,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,31)(26,35,34,44)(27,39,38,43)(28,41,42,45)(32,47,36,48)(33,50,49,40)(37,54,53,55)(46,56,51,52)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=fhi,e^2=i,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, 6681607);