#282041

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(5,6,9,13,10,14,8,12)(7,11,18,20,16,19,17,15)(21,22,24,28,25,29,30,27)(23,26,33,34,31,36,32,35);(2,4)(5,7,8,17,10,16,9,18)(6,11,12,15,14,19,13,20)(21,23,30,32,25,31,24,33)(22,26,27,35,29,36,28,34);(5,6,10,14)(7,15,16,20)(8,13,9,12)(11,18,19,17)(22,27)(23,31)(24,30)(26,34)(28,29)(35,36);(2,4)(7,16)(11,19)(15,20)(17,18)(21,24,25,30)(22,28,29,27)(23,32,31,33)(26,35,36,34)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=hi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,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, 189237);