#438232

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345428);