#124021

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

1650 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:

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

Polycyclic presentation:

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

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, 31217);