#123739

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#385
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30935);