#435322

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 342518);