#435542

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

748 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)(5,6,10,16)(7,13)(8,12,15,11)(9,22)(14,26)(17,32,23,35)(18,29)(19,21)(20,28)(24,30)(25,31)(27,34,36,33);(5,7,12,20,10,18,11,19)(6,13,15,28,16,29,8,21)(9,23,26,33,24,17,25,34)(14,27,30,32,31,36,22,35);(5,8,12,6,10,15,11,16)(7,13,20,28,18,29,19,21)(9,22,26,14,24,30,25,31)(17,27,34,32,23,36,33,35);(2,4)(5,9)(6,14)(7,17)(8,22)(10,24)(11,25)(12,26)(13,27)(15,30)(16,31)(18,23)(19,33)(20,34)(21,35)(28,32)(29,36)

Polycyclic presentation:

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

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, 342738);