#6875534

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

1178 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)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,43,30,35,29,34)(27,32,44,54,37,46,38,47)(28,39,40,33,42,48,41,49)(36,50,52,45,51,56,53,55);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,36)(29,38)(30,37)(31,44)(33,45)(34,47)(35,46)(39,50)(40,52)(41,53)(42,51)(43,54)(48,56)(49,55);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,42)(26,33,35,49)(27,36,37,51)(29,40,31,41)(32,45,46,55)(34,48,43,39)(38,52,44,53)(47,56,54,50);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(27,37)(32,46)(36,51)(38,44)(45,55)(47,54)(50,56)(52,53);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,34,35,43)(27,38,37,44)(28,40,42,41)(32,47,46,54)(33,48,49,39)(36,52,51,53)(45,56,55,50)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6782730);