#6574056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481252);