#479822

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387018);