#322455

order	512
nilpotency class	3
isoclinism	isoclinic class of #322453
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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)(3,10)(4,6)(5,9)(7,12)(8,13)(11,14)(15,16)(17,18,22,27,23,28,32,26)(19,20,29,24,30,25,21,31);(2,6,9,8)(3,11)(7,10,15,14)(12,16)(17,19,22,29,23,30,32,21)(18,24,26,20,28,31,27,25);(1,2)(3,7)(4,8)(5,9)(6,13)(10,12)(11,15)(14,16)(17,20)(18,19)(21,27)(22,31)(23,25)(24,32)(26,29)(28,30);(1,3,4,12,5,11,13,16)(2,7,8,10,9,15,6,14)(17,21,22,19,23,29,32,30)(18,25,26,24,28,20,27,31)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=ghi

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, 229651);