#6732704

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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1178 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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,35)(28,40)(29,33)(30,34)(31,39)(32,41)(36,46)(37,45)(38,44)(42,53)(43,48)(47,54)(49,51)(50,55)(52,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,37)(29,39)(30,36)(32,45)(33,35)(34,44)(38,43)(40,49)(41,51)(42,50)(46,48)(47,55)(52,54)(53,56);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,36)(30,43)(31,44)(34,48)(35,46)(37,50)(38,39)(42,54)(45,55)(47,53)(49,56)(51,52);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,32)(27,37)(29,41)(30,42)(31,45)(33,40)(34,47)(35,49)(36,50)(38,52)(39,51)(43,54)(44,55)(46,56)(48,53);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(21,23)(26,33)(27,38)(28,41)(30,43)(31,44)(35,46)(36,39)(37,50)(45,56)(47,53)(49,55)(51,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6639900);