#110533

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
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

931 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)(11,21)(12,17,27,33)(13,20)(14,18)(15,19)(16,23)(22,35,37,40)(24,30)(25,31)(26,34,28,32)(29,38,39,36);(1,3)(2,5,6,7)(4,8)(9,11,14,24,15,25,13,23)(10,16,20,31,19,30,18,21)(12,22,28,38,27,37,26,36)(17,29,34,40,33,39,32,35);(9,12)(10,17,19,33)(11,22)(13,26)(14,28)(15,27)(16,29,30,39)(18,32,20,34)(21,35,31,40)(23,36)(24,38)(25,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17729);