#426701

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

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

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, 333897);