#341518

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
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

510 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)(9,10)(11,15)(12,16)(13,14)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3,5,8)(2,6,7,4)(9,11,12,13)(10,14,16,15)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(10,14)(11,13)(15,16)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(1,4,5,6)(2,3,7,8)(10,15)(11,13)(14,16)(18,23)(22,25)(24,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248714);