#282334

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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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)(5,6,9,14,10,15,11,16)(7,13,19,20,17,12,8,18)(21,22,26,28,25,29,24,30)(23,27,33,36,31,35,32,34);(2,4)(5,7,11,8,10,17,9,19)(6,12,16,20,15,13,14,18)(21,23,24,32,25,31,26,33)(22,27,30,34,29,35,28,36);(5,8)(6,13)(7,11)(9,17)(10,19)(12,15)(14,18)(16,20)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(2,4)(5,9,10,11)(6,14,15,16)(7,8,17,19)(12,20,13,18)(22,29)(23,31)(28,30)(32,33)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^2=hi,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, 189530);