#6710233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,35)(27,36,39,32)(28,33,42,51)(29,43,31,34)(37,53,55,46)(38,47,44,48)(40,56,54,49)(41,52,45,50);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,39)(26,32,35,36)(28,40,42,54)(29,38,31,44)(33,49,51,56)(34,47,43,48)(37,45,55,41)(46,50,53,52);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,46)(34,50)(35,51)(36,53)(38,54)(39,55)(40,44)(43,52)(47,49)(48,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,30,31)(26,34,35,43)(27,38,39,44)(28,41,42,45)(32,47,36,48)(33,50,51,52)(37,54,55,40)(46,49,53,56);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,38)(28,42)(29,31)(32,48)(33,51)(34,43)(36,47)(37,40)(39,44)(46,49)(53,56)(54,55)

Polycyclic presentation:

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

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, 6617429);