#6815342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814819
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722538);