#273378

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61758	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

861 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)(3,11,14,7)(4,8)(5,12,16,9)(6,10)(13,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,30)(29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,15)(10,11)(18,23)(19,25)(21,31)(24,27)(26,32)(28,30);(1,4,6,15)(2,8,10,13)(3,12,14,9)(5,7,16,11)(17,19,22,18)(20,26,29,24)(21,23,31,25)(27,28,32,30);(1,5,6,16)(2,9,10,12)(3,13,14,8)(4,7,15,11)(17,20)(18,24)(19,26)(21,30)(22,29)(23,27)(25,32)(28,31)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,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, 180574);