#6575739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2153 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)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,25,34)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,40,44)(26,35,30,36)(33,41,43,42);(1,2)(3,4)(6,9)(10,12)(13,15,19,25)(14,20,23,34)(16,24,29,40)(17,30,18,26)(21,33,38,43)(22,36,31,35)(27,39,37,44)(28,42,32,41);(2,4)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,43)(35,39)(36,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,25)(20,34)(24,40)(26,30)(33,43)(35,36)(39,44)(41,42);(2,4)(13,17,19,18)(14,22,23,31)(15,26,25,30)(16,28,29,32)(20,35,34,36)(21,37,38,27)(24,41,40,42)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482935);