#432581

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339777);