#110487

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17683);