#329849

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237045);