#320748

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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)(3,7)(4,5)(6,8)(9,10,12,16,13,11,14,15)(17,18,22,25,21,24,29,32)(19,20,27,28,26,23,31,30);(2,5)(6,7)(9,11,14,16,13,10,12,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,32)(29,31);(9,10)(11,13)(12,15)(14,16)(17,20,21,23)(18,19,24,26)(22,30,29,28)(25,31,32,27);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,15,11,16)(18,24)(19,26)(25,32)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227944);