#282527

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

693 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)(5,6,10,14,8,12,9,13)(7,15,19,16,17,11,18,20)(21,22,25,28)(23,26,30,34)(24,27,31,35)(29,33,32,36);(2,4)(5,7,9,18,8,17,10,19)(6,11,13,16,12,15,14,20)(23,29)(24,31)(26,33)(27,35)(30,32)(34,36);(5,6,8,12)(7,16,17,20)(9,14,10,13)(11,19,15,18)(21,23)(22,26)(24,32)(25,30)(27,36)(28,34)(29,31)(33,35);(2,4)(7,17)(11,15)(16,20)(18,19)(23,30)(26,34)(29,32)(33,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^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,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, 189723);