#317999

order	512
nilpotency class	3
isoclinism	isoclinic class of #317982
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	#19337	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=ehi,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=di,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, 225195);