#290369

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

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, 197565);