#7049255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1207 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)(25,26,30,31)(27,35,36,48)(28,33,40,46)(29,43,44,34)(32,38,45,39)(37,51,52,56)(41,50,42,49)(47,54,55,53);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,40)(34,43)(38,39)(41,42)(47,55)(51,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,30,36,44,38)(26,32,34,35,31,45,43,48)(28,37,42,53,40,52,41,54)(33,47,50,51,46,55,49,56);(2,6)(7,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,37)(29,42)(30,40)(31,46)(32,47)(34,50)(35,51)(36,52)(38,54)(39,53)(41,44)(43,49)(45,55)(48,56);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(25,29,30,44)(26,34,31,43)(27,38,36,39)(28,41,40,42)(32,48,45,35)(33,49,46,50)(37,53,52,54)(47,51,55,56)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgi,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6956451);