#262406

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59652	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1027 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:

(2,3)(5,6)(7,13)(8,10)(9,17)(11,15)(12,16)(14,20)(18,19)(21,22,24,28)(23,30,32,26)(25,27,34,29)(31,36,33,35);(1,2,4,3)(6,10)(9,16)(13,20)(15,18)(21,23,24,32)(22,26,28,30)(25,31,34,33)(27,35,29,36);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)(22,27)(23,31)(25,34)(28,29)(32,33)(35,36);(1,3,4,2)(6,10)(11,19)(12,17)(13,20)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169602);