#352399

order	512
nilpotency class	3
isoclinism	isoclinic class of #352384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

632 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)(3,10,12,7)(4,14,9,6)(11,16,15,13)(17,18)(19,21)(20,27)(22,26)(23,25)(24,31)(28,30)(29,32);(1,3,5,12)(2,7,8,10)(4,15,9,11)(6,13,14,16)(17,18,23,19,22,26,21,25)(20,28,31,29,32,24,30,27);(1,4,6,2,5,9,14,8)(3,11,13,10,12,15,16,7)(17,19,23,26,22,25,21,18)(20,29,31,24,32,27,30,28);(2,8)(3,12)(4,9)(13,16)(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,29)(26,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=di,d^c=di,g^d=gh

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, 259595);