#317967

order	512
nilpotency class	3
isoclinism	isoclinic class of #317966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225163);