#467238

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

975 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=g,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 374434);