#426742

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=ef,f^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^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, 333938);