#330338

order	512
nilpotency class	3
isoclinism	isoclinic class of #329707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

656 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)(9,10,13,14,12,15,11,16)(17,18,23,31,22,20,21,25)(19,24,28,26,29,30,27,32);(2,5)(6,8)(17,19,22,29)(18,24,20,30)(21,27,23,28)(25,32,31,26);(3,7)(6,8)(10,14)(11,13)(15,16)(17,20,22,18)(19,26,29,32)(21,25,23,31)(24,27,30,28);(1,3,4,7)(2,6,5,8)(10,15)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237534);