#273380

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,e^d=ei,f^d=fhi

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, 180576);