#396110

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=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, 303306);