#417145

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,24)(13,27)(14,17)(15,22)(16,26)(18,20)(21,37)(23,35)(25,32)(28,33)(29,38)(30,36)(31,40)(34,39);(2,5)(6,7)(9,11,20,36,14,23,13,22)(10,15,27,35,17,30,18,19)(12,21,28,40,26,39,25,38)(16,31,33,37,24,29,32,34);(9,12)(10,16,17,24)(11,21,23,39)(13,28)(14,26)(15,29)(18,33,27,32)(19,34)(20,25)(22,40,36,38)(30,31)(35,37);(1,3,4,8)(2,6,5,7)(10,17)(15,30)(16,24)(18,27)(19,35)(29,31)(32,33)(34,37)

Polycyclic presentation:

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

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, 324341);