#341616

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248812);