#7111419

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,44)(30,33)(31,40)(34,36)(37,47)(38,46)(39,53)(41,56)(42,51)(43,52)(45,50)(48,54)(49,55);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,36,54,30,39,29,40)(26,31,44,53,33,48,34,35)(28,41,45,38,42,37,43,55)(32,49,52,47,51,46,50,56);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,42)(26,32,33,51)(27,37,39,41)(29,43,36,45)(31,46,48,49)(34,50,44,52)(35,47,53,56)(38,54,55,40);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,32)(27,38)(29,45)(30,42)(31,47)(33,51)(34,52)(35,46)(36,43)(37,40)(39,55)(41,54)(44,50)(48,56)(49,53);(10,16)(11,18)(15,24)(17,19)(26,33)(27,39)(28,42)(34,44)(40,54)(43,45)(46,49)(47,56)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=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, 7018615);