#6539909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,31)(26,34)(27,36)(29,35)(32,38)(33,37)(39,41)(40,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,28,19,29,18,27)(14,20,23,25,24,36,22,35)(16,26,32,40,33,42,31,41)(21,34,37,39,38,44,30,43);(2,4)(6,9)(10,12)(13,16,17,32,19,33,18,31)(14,21,23,37,24,38,22,30)(15,26,28,40,29,42,27,41)(20,34,25,39,36,44,35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,37,38,30)(26,40,42,41)(34,43,44,39);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,33,31)(20,25,36,35)(21,30,38,37)(26,41,42,40)(34,39,44,43)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh

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, 6447105);