#6710228

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fg,e^2=f,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, 6617424);