#6532307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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

1306 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)(25,26,31,44,30,34,29,35)(27,36,43,32,38,51,39,47)(28,33,45,56,41,49,42,50)(37,52,55,46,40,54,53,48);(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)(26,32)(28,40)(29,43)(30,38)(31,39)(33,48)(34,47)(35,51)(36,44)(37,41)(42,53)(45,55)(46,49)(50,52)(54,56);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,41)(26,33,34,49)(27,37,38,40)(29,42,31,45)(32,46,47,48)(35,50,44,56)(36,52,51,54)(39,53,43,55);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(28,41)(33,49)(37,40)(42,45)(46,48)(50,56)(52,54)(53,55);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(26,34)(27,38)(28,41)(35,44)(39,43)(42,45)(46,48)(52,54)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439503);