#413342

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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)(7,17,20,26)(8,21)(9,14)(10,15)(11,16)(12,18,28,25)(13,24)(19,27,34,36)(22,30)(23,29)(31,33,35,32);(1,2)(3,4)(7,18,20,25)(8,22)(12,26,28,17)(13,29)(19,33,34,32)(21,30)(23,24)(27,31,36,35);(2,4)(5,7,9,25,11,20,10,18)(6,12,14,26,16,28,15,17)(8,19,23,33,22,34,24,32)(13,27,21,35,29,36,30,31);(5,8,10,24,11,22,9,23)(6,13,14,21,16,29,15,30)(7,19,18,32,20,34,25,33)(12,27,26,35,28,36,17,31)

Polycyclic presentation:

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

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, 320538);