#461618

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
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

967 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)(5,6,10,15,9,16,11,14)(7,17,21,12,20,27,18,29)(8,22,25,30,24,13,26,31)(19,32,34,28,23,36,33,35);(7,18)(10,11)(12,27)(14,15)(17,29)(19,33)(20,21)(23,34)(25,26)(28,36)(30,31)(32,35);(1,2,3,4)(5,7,10,21,9,20,11,18)(6,12,14,17,16,29,15,27)(8,23,25,33,24,19,26,34)(13,28,30,32,22,35,31,36);(5,8,9,24)(6,13)(7,19,20,23)(10,25,11,26)(12,28)(14,30)(15,31)(16,22)(17,32)(18,33,21,34)(27,36)(29,35)

Polycyclic presentation:

2^9|a^2=eg,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,h^c=hi,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, 368814);