#400506

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,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, 307702);