#231474

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂³	#518
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138670);