#374169

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281365);