#341609

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=hi,d^2=e,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, 248805);