#124019

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	#379
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1674 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)(5,6,7,8)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,26)(20,29)(22,28)(23,30)(24,37)(27,34)(31,36)(32,35)(33,38)(39,40);(9,11,13,22,14,21,26,35)(10,15,18,28,17,19,25,32)(12,20,24,33,23,34,36,40)(16,29,31,39,30,27,37,38);(1,2,3,4)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,31)(19,29)(21,34)(22,33)(25,37)(26,36)(28,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,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, 31215);