#462322

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

880 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)(5,6,9,15,10,14,25,12)(7,16,19,27,20,11,17,28)(8,21,23,30,22,29,24,13)(18,33,34,26,35,36,32,31);(5,7)(6,11)(8,18)(9,17)(10,20)(12,28)(13,26)(14,16)(15,27)(19,25)(21,33)(22,35)(23,32)(24,34)(29,36)(30,31);(6,12)(7,17)(8,22)(9,25)(13,29)(14,15)(18,34)(19,20)(21,30)(27,28)(32,35)(33,36);(2,4)(5,8,9,23,10,22,25,24)(6,13,15,21,14,30,12,29)(7,18,17,32,20,35,19,34)(11,26,27,33,16,31,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369518);