#478426

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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)(5,6,11,14,9,15,10,25)(7,16,20,29,19,12,17,27)(8,21,26,31,23,13,24,30)(18,32,33,35,22,28,34,36);(5,7,9,19)(6,12)(8,22,23,18)(10,17,11,20)(13,28)(14,27)(15,16)(21,32)(24,33,26,34)(25,29)(30,36)(31,35);(7,17)(10,11)(12,27)(14,25)(16,29)(18,34)(19,20)(22,33)(24,26)(28,36)(30,31)(32,35);(2,4)(5,8,9,23)(6,13,15,21)(7,18,19,22)(10,26,11,24)(12,28,16,32)(14,31,25,30)(17,33,20,34)(27,35,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385622);