#330413

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,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, 237609);