#281727

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

711 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)(5,6,9,14,11,16,10,15)(7,12,17,13,8,19,18,20)(21,22,27,33,25,24,26,29)(23,30,34,35,31,28,32,36);(2,4)(5,7,9,17,11,8,10,18)(6,12,14,13,16,19,15,20)(21,23,26,32,25,31,27,34)(22,28,29,35,24,30,33,36);(5,8,11,7)(6,13,16,20)(9,17,10,18)(12,15,19,14)(21,24,25,22)(23,30,31,28)(26,29,27,33)(32,35,34,36);(2,4)(22,24)(28,30)(29,33)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188923);