#467782

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

889 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:

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

Polycyclic presentation:

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

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, 374978);