#282108

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(5,6,9,12,10,13,8,18)(7,11,16,19,17,20,15,14)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(2,4)(5,7,9,16,10,17,8,15)(6,11,12,19,13,20,18,14)(21,23,24,30)(22,26,27,34)(25,29,32,31)(28,36,35,33);(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,20)(17,19)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,20,14)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 189304);