#6709996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1206 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,44)(36,51)(37,45)(38,46)(39,47)(40,48)(41,49)(42,52)(43,50)(53,55)(54,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,38)(26,31,33,46)(28,39,40,54)(29,36,43,37)(32,47,48,56)(34,51,50,45)(35,42,53,41)(44,52,55,49);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,35)(29,42)(30,40)(31,44)(33,48)(34,52)(36,39)(37,54)(38,53)(41,43)(45,56)(46,55)(47,51)(49,50);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,40)(32,48)(35,53)(39,54)(41,42)(44,55)(47,56)(49,52);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,36)(29,43)(31,45)(32,48)(35,39)(37,38)(41,42)(44,56)(46,51)(47,55)(53,54)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6617192);