#361863

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269059);