#289321

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efh,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196517);