#374167

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	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1601 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)(11,15)(12,16,24,30)(13,17)(14,18)(19,32,22,26)(20,27)(21,28)(23,29,36,39)(25,31)(33,40,35,37)(34,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,26)(17,27)(18,28)(19,24)(23,35)(25,34)(29,37)(30,32)(31,38)(33,36)(39,40);(9,12)(10,16,18,30)(11,19,21,22)(13,23)(14,24)(15,26)(17,29,31,39)(20,33,34,35)(25,36)(27,37)(28,32)(38,40);(2,6)(5,7)(9,13,14,25)(10,17,18,31)(11,20,21,34)(12,23,24,36)(15,27,28,38)(16,29,30,39)(19,33,22,35)(26,37,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^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, 281363);