#6520415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,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, 6427611);