#6709331

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

1262 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)(27,36)(28,40)(29,44)(30,35)(31,34)(32,39)(33,43)(37,47)(38,48)(41,50)(42,51)(45,49)(46,52)(53,55)(54,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,39)(26,32,35,36)(28,41,43,54)(29,38,31,45)(33,50,40,56)(34,48,44,49)(37,46,55,42)(47,51,53,52);(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)(29,42)(30,43)(31,46)(32,47)(34,51)(35,40)(36,53)(38,54)(39,55)(41,45)(44,52)(48,50)(49,56);(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)(26,34,35,44)(27,38,39,45)(28,42,43,46)(32,48,36,49)(33,51,40,52)(37,54,55,41)(47,50,53,56);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,35)(27,38)(28,43)(29,31)(32,48)(36,49)(37,41)(39,45)(47,56)(50,53)(51,52)(54,55)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6616527);