#386020

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35634	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1659 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,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, 293216);