#445429

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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)(9,10,15,20)(11,21,25,27)(12,24,16,17)(13,28)(14,22,30,19)(18,32)(23,35)(26,40)(29,34)(31,36)(33,37)(38,39);(3,7)(5,8)(9,11,14,24,15,25,30,17)(10,16,19,21,20,12,22,27)(13,29,31,39,32,40,33,23)(18,35,36,26,28,38,37,34);(1,3,4,7)(2,5,6,8)(9,12,15,16)(10,17,20,24)(11,22,25,19)(13,26,32,34)(14,27,30,21)(18,29,28,40)(23,36,39,37)(31,38,33,35);(9,13,30,31,15,32,14,33)(10,18,22,36,20,28,19,37)(11,23,17,29,25,39,24,40)(12,26,21,38,16,34,27,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=gi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352625);