#281720

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 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)(5,6,10,12,9,13,8,14)(7,15,19,16,18,11,17,20)(21,22,26,28,25,30,24,29)(23,27,33,36,31,35,32,34);(2,4)(5,7,8,17,9,18,10,19)(6,11,14,16,13,15,12,20)(21,23,26,33,25,31,24,32)(22,27,28,36,30,35,29,34);(5,6,9,13)(7,16,18,20)(8,12,10,14)(11,19,15,17)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(2,4)(5,8,9,10)(6,12,13,14)(7,17,18,19)(11,20,15,16)(21,24,25,26)(22,29,30,28)(23,32,31,33)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=hi,d^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, 188916);