#6542724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6449920);