#6491822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24402	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^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=dh,g^c=gi,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, 6399018);