#6575025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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)(7,10)(8,9)(11,12)(15,24)(20,25)(26,28)(27,41)(30,43)(35,40)(37,42)(38,44);(6,9)(10,12)(13,14,18,23)(15,25,24,20)(16,29,32,36)(17,19,33,22)(21,31,39,34)(26,40,28,35)(27,38,41,44)(30,37,43,42);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,26)(17,33)(21,36)(22,23)(24,28)(27,30)(29,39)(31,34)(35,40)(38,44)(41,43);(13,15,18,24)(14,20,23,25)(16,30,32,43)(17,26,33,28)(19,35,22,40)(21,38,39,44)(27,34,41,31)(29,42,36,37);(13,16,18,32)(14,21,23,39)(15,27,24,41)(17,34,33,31)(19,36,22,29)(20,37,25,42)(26,30,28,43)(35,44,40,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482221);