#6497452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1396 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,37)(28,39)(29,36)(30,35)(31,34)(32,38)(33,40);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,29,38,31,39,30,33)(26,32,35,37,34,40,36,28);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,29,32,31,37,30,40)(26,33,35,27,34,38,36,39);(11,17)(14,21)(18,23)(22,24)(25,26)(27,32)(28,33)(29,35)(30,36)(31,34)(37,38)(39,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,34)(28,37)(32,40)(35,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404648);