#329379

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

731 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:

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

Polycyclic presentation:

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

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, 236575);