#6575771

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2148 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)(13,14,19,24)(15,20,26,34)(16,28,30,38)(17,23,18,22)(21,32,36,29)(25,39,40,44)(27,37,31,35)(33,41,43,42);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,24,34)(16,25,30,40)(17,31,18,27)(21,33,36,43)(22,37,23,35)(28,39,38,44)(29,42,32,41);(2,4)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,33)(22,28)(23,38)(24,36)(26,40)(27,41)(31,42)(34,43)(35,44)(37,39);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,26)(20,34)(25,40)(27,31)(33,43)(35,37)(39,44)(41,42);(2,4)(13,17,19,18)(14,22,24,23)(15,27,26,31)(16,29,30,32)(20,35,34,37)(21,28,36,38)(25,41,40,42)(33,44,43,39)

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,e^a=ei,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482967);