#340703

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 247899);