#6578561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485757);