#6528413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=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, 6435609);