#492588

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
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

478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#509, #72342600, #72342604
maximal normal subgroup of	SD₁₆×#509, #72342600, #72342604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399784);