#383752

order	512
nilpotency class	3
isoclinism	isoclinic class of #213818
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290948);