#7186326

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(27,35)(28,39)(29,34)(30,36)(31,32)(33,41)(37,48)(38,49)(40,52)(42,45)(43,46)(44,50)(47,53)(51,56)(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)(25,27,30,31)(26,32,36,35)(28,37,42,41)(29,43,44,38)(33,45,48,39)(34,49,50,46)(40,53,54,51)(47,52,56,55);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,28,31,41,30,42,27,37)(26,33,35,39,36,48,32,45)(29,40,43,53,44,54,38,51)(34,47,49,52,50,56,46,55);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,29)(26,34)(27,38)(28,40)(30,44)(31,43)(32,46)(33,47)(35,49)(36,50)(37,51)(39,52)(41,53)(42,54)(45,55)(48,56);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(25,27,30,31)(26,35,36,32)(28,41,42,37)(29,43,44,38)(33,45,48,39)(34,46,50,49)(40,51,54,53)(47,52,56,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fgi,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^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, 7093522);