#6787894

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6695090);