#6744861

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1199 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)(3,7)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,36)(28,33)(29,43)(30,35)(31,34)(32,39)(37,53)(38,47)(40,56)(41,52)(42,51)(44,48)(45,50)(46,55)(49,54);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,40)(29,38)(30,39)(31,44)(33,49)(34,47)(35,36)(37,41)(42,54)(43,48)(45,55)(46,52)(50,53)(51,56);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,42)(26,33,35,51)(27,37,39,55)(29,41,31,45)(32,46,36,53)(34,50,43,52)(38,54,44,40)(47,49,48,56);(1,3,5,4)(2,7,6,8)(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);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,38)(29,31)(32,48)(34,43)(36,47)(37,54)(39,44)(40,55)(41,45)(46,56)(49,53)(50,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6652057);