#6576912

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

2682 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)(13,14,17,21)(15,23,25,19)(16,27,29,38)(18,32,31,22)(20,33,37,30)(24,40,28,44)(26,35,42,39)(34,41,36,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,28)(17,25)(18,26)(20,36)(21,23)(22,35)(24,29)(27,40)(30,43)(31,42)(32,39)(33,41)(34,37)(38,44);(2,4)(13,16,17,29)(14,20,21,37)(15,24,25,28)(18,33,31,30)(19,34,23,36)(22,38,32,27)(26,43,42,41)(35,40,39,44);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(16,29)(20,37)(24,28)(27,38)(30,33)(34,36)(40,44)(41,43);(2,4)(15,25)(19,23)(24,28)(26,42)(34,36)(35,39)(40,44)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,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, 6484108);