#492595

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	#33838	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

490 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)(9,10,13,15)(11,14,20,22)(12,21,19,17)(16,24,23,18);(1,2,5,7)(3,6,4,8);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(1,3)(4,5)(6,8)(10,15)(14,22)(16,23)(17,21)

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,d^a=dgi,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

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#509, #72342594, #72342602
maximal normal subgroup of	SD₁₆×#509, #72342594, #72342602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399791);