#6709711

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

1211 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)(25,26,30,35)(27,36,39,32)(28,40,43,33)(29,44,31,34)(37,47,55,53)(38,48,45,49)(41,50,54,56)(42,51,46,52);(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)(25,28,30,43)(26,33,35,40)(27,37,39,55)(29,42,31,46)(32,47,36,53)(34,51,44,52)(38,54,45,41)(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)(27,38)(28,43)(29,31)(32,49)(33,40)(34,44)(36,48)(37,41)(39,45)(47,50)(53,56)(54,55)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgi,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, 6616907);