#282369

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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)(5,6,8,13,10,12,9,14)(7,15,17,20,19,16,18,11)(21,22,25,33,27,24,26,29)(23,28,34,35,32,30,31,36);(2,4)(5,6,8,13,10,12,9,14)(7,16,17,11,19,15,18,20)(21,23,26,31,27,32,25,34)(22,28,29,36,24,30,33,35);(5,7,10,19)(6,11,12,20)(8,18,9,17)(13,16,14,15)(21,24,27,22)(23,30,32,28)(25,33,26,29)(31,36,34,35);(2,4)(6,12)(11,20)(13,14)(15,16)(21,25,27,26)(22,29,24,33)(23,31,32,34)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189565);