#7124820

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

1365 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 7032016);