#206031

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
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×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 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)(3,5,7,8)(4,6)(9,10,14,18,15,19,13,29)(11,20,24,31,22,34,23,16)(12,25,27,32,28,39,26,17)(21,35,38,40,36,30,37,33);(1,3)(2,5,6,8)(4,7)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,32,35)(18,20,29,34)(25,33,39,40)(26,37)(27,38)(28,36);(2,6)(5,8)(9,12,13,26,15,28,14,27)(10,17,18,25,19,32,29,39)(11,21,23,37,22,36,24,38)(16,30,20,33,31,35,34,40);(3,7)(5,8)(11,22)(16,31)(20,34)(21,36)(23,24)(30,35)(33,40)(37,38)

Polycyclic presentation:

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

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, 113227);