#329998

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

576 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)(9,10,11,15,12,16,13,14)(17,18,22,31,20,24,21,25)(19,26,30,27,28,23,29,32);(1,2,4,5)(3,6,7,8)(9,11,12,13)(10,14,16,15)(17,19,20,28)(18,23,24,26)(21,29,22,30)(25,27,31,32);(1,3)(2,6)(4,7)(5,8)(10,15)(11,13)(14,16)(17,18,20,24)(19,27,28,32)(21,31,22,25)(23,30,26,29);(3,7)(6,8)(9,11,12,13)(10,15,16,14)(18,24)(23,26)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,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, 237194);