#6532254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(4,8)(5,7)(25,26,29,33,30,34,44,51)(27,35,38,46,37,50,43,31)(28,39,42,55,41,32,47,49)(36,53,54,45,40,52,56,48);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,40)(29,43)(30,37)(32,48)(33,50)(34,46)(35,51)(36,41)(38,44)(39,45)(42,54)(47,56)(49,52)(53,55);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,41)(26,32,34,39)(27,36,37,40)(29,42,44,47)(31,45,46,48)(33,49,51,55)(35,52,50,53)(38,54,43,56);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(28,41)(32,39)(36,40)(42,47)(45,48)(49,55)(52,53)(54,56);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(27,37)(28,41)(31,46)(32,39)(35,50)(38,43)(42,47)(49,55)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439450);