#7187133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(12,20)(15,21)(19,24)(22,23)(25,26)(27,32)(28,39)(29,34)(30,35)(31,36)(33,40)(37,48)(38,45)(41,52)(42,46)(43,50)(44,49)(47,53)(51,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,36,35)(28,40,42,37)(29,38,44,43)(33,46,48,39)(34,45,49,50)(41,53,54,51)(47,55,56,52);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,40,31,42,27,37)(26,33,32,39,36,48,35,46)(29,41,43,53,44,54,38,51)(34,47,45,52,49,56,50,55);(1,4,5,2)(3,6,8,7)(11,18)(14,17)(19,24)(22,23)(25,29,31,44)(26,34,36,49)(27,38,30,43)(28,41,42,54)(32,45,35,50)(33,47,48,56)(37,51,40,53)(39,52,46,55);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,31,30)(26,35,36,32)(28,37,42,40)(29,43,44,38)(33,46,48,39)(34,45,49,50)(41,53,54,51)(47,52,56,55)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fg,d^2=fi,e^2=h,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^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094329);