#7039919

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

1187 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=f,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, 6947115);