#238620

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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

888 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145816);