#492460

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
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	#3391	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

427 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)(7,13)(8,15)(9,16)(10,14)(17,18,19,22)(20,23,26,30)(21,29,27,25)(24,32,31,28);(1,2,5,7)(3,9,11,15)(4,13,12,6)(8,14,16,10)(18,22)(23,30)(24,31)(25,29);(2,7)(6,13)(8,16)(9,15)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(1,3,5,11)(2,8,7,16)(4,14,12,10)(6,15,13,9)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399656);