#6576910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#32885	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2668 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484106);