#7173198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080394);