#273347

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	#3488	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

970 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)(5,8)(7,13)(10,15,16,14)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,30)(29,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,23)(19,25)(21,31)(24,27)(26,32)(28,30);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,19,22,18)(20,26,29,24)(21,23,31,25)(27,28,32,30);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,27)(25,32)(28,31)

Polycyclic presentation:

2^9|a^2=fh,c^2=i,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, 180543);