#6509715

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,33)(28,42)(29,35)(30,36)(31,46)(32,37)(34,44)(38,48)(39,49)(40,55)(41,50)(43,56)(45,52)(47,53)(51,54);(2,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,33)(28,43)(29,38)(30,39)(31,40)(32,41)(34,51)(35,48)(36,49)(37,50)(42,56)(44,54)(45,53)(46,55)(47,52);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,34)(27,38)(29,32)(30,44)(31,45)(33,47)(35,36)(37,52)(39,53)(40,54)(41,43)(42,46)(48,55)(49,56)(50,51);(1,4,5,3)(2,8,6,7)(12,20)(15,24)(17,19)(21,23)(25,29,32,28)(26,35,37,42)(27,38,41,43)(30,44,31,45)(33,48,50,56)(34,46,52,36)(39,54,40,53)(47,49,51,55);(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,36,37,46)(27,39,41,40)(28,44,29,45)(33,49,50,55)(34,35,52,42)(38,53,43,54)(47,56,51,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,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, 6416911);