#322004

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229200);