#282328

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

709 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)(5,6,9,15,11,16,10,14)(7,12,17,20,18,13,8,19)(21,22,24,28,25,29,30,27)(23,26,32,34,33,35,31,36);(2,4)(5,7,9,17,11,18,10,8)(6,12,15,20,16,13,14,19)(21,23,30,31,25,33,24,32)(22,26,27,36,29,35,28,34);(5,8,11,17)(6,13,16,12)(7,9,18,10)(14,19,15,20)(21,22)(23,26)(24,27)(25,29)(28,30)(31,34)(32,36)(33,35);(2,4)(5,9,11,10)(6,14,16,15)(7,8,18,17)(12,20,13,19)(21,24,25,30)(22,27,29,28)(23,31,33,32)(26,34,35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,d^2=h,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, 189524);