#445427

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

Polycyclic presentation:

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

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, 352623);