#492153

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
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	C₂×D₁₆	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 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)(17,18,20,23)(19,25,22,30)(21,24,28,31)(26,27,32,29);(1,2,5,8)(3,9,11,15)(4,13,12,6)(7,14,16,10)(18,23)(24,31)(25,30)(26,32);(18,24)(19,22)(21,29)(23,31)(25,26)(27,28)(30,32);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(17,19,20,22)(18,25,23,30)(21,27,28,29)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,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=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319599
maximal normal subgroup of	#72319599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399349);