#282537

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24658	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=hi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 189733);