#6577927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1959 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)(8,12)(9,10);(6,10)(11,12)(13,14,18,23)(15,24,27,39)(16,28,32,20)(17,21,34,22)(19,33,36,26)(25,37,43,41)(29,35,42,44)(30,40,31,38);(2,4)(13,15,18,27)(14,19,23,36)(16,29,32,42)(17,33,34,26)(20,37,28,41)(21,24,22,39)(25,30,43,31)(35,40,44,38);(5,7,9,8)(6,11,10,12)(13,16,18,32)(14,20,23,28)(15,25,27,43)(17,30,34,31)(19,35,36,44)(21,38,22,40)(24,41,39,37)(26,42,33,29);(5,8,9,7)(6,12,10,11)(14,21)(16,30)(17,34)(19,24)(22,23)(25,42)(26,33)(29,43)(31,32)(35,44)(36,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485123);