#123969

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1638 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)(6,8)(9,10,14,19,15,18,13,26)(11,16,22,31,21,32,28,30)(12,17,25,35,24,36,27,34)(20,29,37,33,38,40,23,39);(2,4)(9,11)(10,16,18,32)(12,23,24,37)(13,22)(14,28)(15,21)(17,33)(19,30,26,31)(20,25,38,27)(29,35)(34,40)(36,39);(5,6,7,8)(9,12)(10,17)(11,20)(13,27)(14,25)(15,24)(16,29)(18,36)(19,35)(21,38)(22,37)(23,28)(26,34)(30,39)(31,33)(32,40)

Polycyclic presentation:

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

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, 31165);