#7186180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(11,17,18,14)(12,20,22,15)(19,23,21,24)(25,26)(27,34)(28,39)(29,33)(30,35)(31,38)(32,36)(37,48)(40,52)(41,46)(42,44)(43,49)(45,50)(47,51)(53,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,17)(12,21,22,19)(15,24,20,23)(25,27,30,38)(26,31,35,34)(28,36,41,51)(29,42,43,37)(32,39,47,46)(33,48,49,44)(40,53,54,50)(45,55,56,52);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,28,38,51,30,41,27,36)(26,32,34,46,35,47,31,39)(29,40,42,53,43,54,37,50)(33,45,48,55,49,56,44,52);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,29)(26,33)(27,37)(28,40)(30,43)(31,44)(32,45)(34,48)(35,49)(36,50)(38,42)(39,52)(41,54)(46,55)(47,56)(51,53);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(25,27,30,38)(26,34,35,31)(28,36,41,51)(29,42,43,37)(32,46,47,39)(33,44,49,48)(40,53,54,50)(45,52,56,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fg,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093376);