#268620

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=i,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175816);