#282356

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,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, 189552);