#444920

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=eg,d^2=e,e^2=i,f^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,f^c=fi

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, 352116);