#329618

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
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

710 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=h,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, 236814);