#6819948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1139 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fgh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727144);