#273320

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	#3494	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

846 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)(19,26)(20,22)(21,25)(23,27)(24,30)(28,31)(29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,15)(10,11)(18,22)(20,30)(23,26)(24,25)(27,28)(31,32);(1,4,6,15)(2,8,10,13)(3,12,14,9)(5,7,16,11)(17,18)(19,23)(20,24)(21,25)(22,30)(26,27)(28,32)(29,31);(1,5,6,16)(2,9,10,12)(3,13,14,8)(4,7,15,11)(17,19,21,29)(18,23,25,31)(20,28,30,27)(22,26,24,32)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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, 180516);