#460099

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 367295);