#6577011

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	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2662 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)(13,14,19,23)(15,20,25,35)(16,27,29,40)(17,22,18,32)(21,33,39,30)(24,41,28,44)(26,36,31,37)(34,42,38,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,28)(17,31)(18,26)(19,25)(21,38)(22,37)(23,35)(24,29)(27,44)(30,42)(32,36)(33,43)(34,39)(40,41);(2,4)(13,16)(14,21)(15,24)(17,30)(18,33)(19,29)(20,34)(22,40)(23,39)(25,28)(26,42)(27,32)(31,43)(35,38)(36,41)(37,44);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(15,25)(16,29)(20,35)(21,39)(26,31)(27,40)(30,33)(36,37);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,31)(16,30,29,33)(20,36,35,37)(21,40,39,27)(24,42,28,43)(34,41,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6484207);