#492113

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
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	#2206	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(4,12)(7,15)(8,13)(9,16)(10,14)(17,18)(19,24)(20,23)(21,30)(22,26)(25,28)(27,31)(29,32);(1,2,5,8)(3,9,11,15)(4,13,12,6)(7,14,16,10);(18,23)(19,21)(20,28)(24,25)(26,32)(27,29)(30,31);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(17,19,22,21)(18,24,26,30)(20,27,29,28)(23,31,32,25)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,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,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320531
maximal normal subgroup of	#72320531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399309);