#6789070

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

1189 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)(25,26,30,32)(27,36,37,48)(28,40,42,34)(29,35,31,45)(33,43,46,39)(38,53,52,51)(41,50,44,49)(47,55,56,54);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(26,32)(27,37)(34,40)(35,45)(38,52)(39,43)(49,50)(54,55);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,27,29,43,30,37,31,39)(26,33,45,48,32,46,35,36)(28,38,44,54,42,52,41,55)(34,47,49,53,40,56,50,51);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,41)(32,40)(33,47)(35,50)(36,51)(37,52)(39,55)(43,54)(45,49)(46,56)(48,53);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,32,45)(27,39,37,43)(28,41,42,44)(33,48,46,36)(34,49,40,50)(38,54,52,55)(47,51,56,53)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6696266);