#457759

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

798 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)(5,6,11,17,10,16,12,18)(7,9,20,28,15,13,21,29)(8,22,25,31,27,32,26,14)(19,33,34,30,23,36,35,24);(5,7,10,15)(6,13,16,9)(8,23)(11,20,12,21)(14,30)(17,29,18,28)(19,27)(22,36)(24,31)(25,35)(26,34)(32,33);(5,8,12,25,10,27,11,26)(6,14,18,22,16,31,17,32)(7,19,21,34,15,23,20,35)(9,24,29,33,13,30,28,36);(2,4)(5,9,10,13)(6,15,16,7)(8,24,27,30)(11,29,12,28)(14,23,31,19)(17,20,18,21)(22,34,32,35)(25,36,26,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364955);