#451509

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

639 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)(5,6,11,15)(7,16,14,10)(8,12)(9,24,26,18)(13,29,30,25)(17,21)(19,22)(20,33)(23,28)(27,34)(31,35)(32,36);(5,7,16,6,11,14,10,15)(8,19,23,17,22,12,21,28)(9,13,29,24,26,30,25,18)(20,31,36,27,35,33,34,32);(5,8,16,23,11,22,10,21)(6,12,14,28,15,19,7,17)(9,20,29,36,26,35,25,34)(13,27,24,33,30,32,18,31);(2,4)(5,9,10,25,11,26,16,29)(6,13,14,24,15,30,7,18)(8,20,21,34,22,35,23,36)(12,27,28,33,19,32,17,31)

Polycyclic presentation:

2^9|a^2=gh,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, 358705);