#273389

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
prime radical	#3536	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,29,33,38)(20,34,36,42)(21,27)(22,25)(23,28)(24,30,40,32)(26,35,41,37)(31,39)(43,44)(45,47)(46,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,24)(20,31)(21,32)(22,33)(23,30)(25,40)(26,39)(27,38)(28,29)(34,43)(35,44)(36,46)(37,45)(41,48)(42,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(18,25)(19,30)(21,23)(24,38)(26,41)(29,40)(31,44)(32,33)(35,37)(39,47)(43,48)(45,46);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,20,22,36)(18,26,25,41)(19,31,33,46)(21,37,23,35)(24,39,40,48)(27,42,28,34)(29,43,38,47)(30,44,32,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^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, 180585);