#329644

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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)(9,10,14,16,13,11,12,15)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32);(2,5)(6,8)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,14,16)(18,24)(19,27)(20,22)(23,31)(26,32)(28,29);(3,7)(6,8)(17,20,21,22)(18,25,24,30)(19,28,29,27)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=i,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, 236840);