#352365

order	512
nilpotency class	3
isoclinism	isoclinic class of #352352
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

656 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,24)(19,26)(20,28)(21,22)(25,32)(30,31);(3,10)(6,15)(9,16)(11,14)(17,18,21,24,23,26,22,19)(20,25,28,29,31,32,30,27);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,21,18,23,24,22,26)(20,29,28,32,31,27,30,25);(3,10)(6,15)(9,16)(11,14)(17,20)(18,25)(19,27)(21,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,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^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, 259561);