#362831

order	512
nilpotency class	3
isoclinism	isoclinic class of #214656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,21,23)(11,17,19,24,18,14,20,22);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,20)(13,18)(15,17)(16,24)(19,21)(22,23);(9,11)(10,14,15,17)(12,19)(13,18)(16,22,23,24)(20,21);(2,6)(5,7)(10,15)(12,21)(14,17)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270027);