#382812

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1346 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,30)(23,31)(24,32)(25,33)(29,35)(34,38)(36,39)(37,40);(9,11,14,21)(10,15,18,28)(12,19,23,35)(13,20,25,37)(16,29,31,26)(17,27,33,40)(22,34,24,36)(30,39,32,38);(2,5)(6,8)(9,12)(10,16)(11,19)(13,24,25,22)(14,23)(15,26)(17,32,33,30)(18,31)(20,36,37,34)(21,35)(27,39,40,38)(28,29);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,30)(18,33)(19,34)(21,37)(23,24)(26,38)(28,40)(29,39)(31,32)(35,36)

Polycyclic presentation:

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

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, 290008);