#491784

order	512
nilpotency class	4
isoclinism	isoclinic class of #491712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	D₄×D₈	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342446, SD₃₂×AGL(1,ℤ/8), #72342602
maximal normal subgroup of	#72342446, SD₃₂×AGL(1,ℤ/8), #72342602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398980);