#458404

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

752 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)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,34)(20,29)(21,28)(22,36)(23,33)(24,31)(25,32)(26,27)(30,35);(5,7,12,21,11,20,10,9)(6,13,18,29,17,28,16,15)(8,19,23,35,25,22,24,26)(14,30,31,34,32,27,33,36);(5,8,12,23,11,25,10,24)(6,14,18,31,17,32,16,33)(7,19,21,35,20,22,9,26)(13,27,29,36,28,30,15,34);(2,4)(5,9,10,20,11,21,12,7)(6,15,16,28,17,29,18,13)(8,22,23,26,25,19,24,35)(14,30,31,34,32,27,33,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=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, 365600);