#267312

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1006 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)(5,6,10,12,9,14,8,13)(7,15,16,11,18,20,17,19)(21,22,24,28,26,30,25,29)(23,27,32,35,33,36,31,34);(2,4)(21,23,26,33)(22,27,30,36)(24,32,25,31)(28,35,29,34);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(22,28)(23,31)(24,25)(29,30)(32,33)(34,35);(2,4)(5,8,9,10)(6,12,14,13)(7,16,18,17)(11,15,19,20)(21,24,26,25)(22,29,30,28)(23,32,33,31)(27,34,36,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 174508);