#7040682

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

1183 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=fi,e^2=fi,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=dhi,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, 6947878);