#253475

order	512
nilpotency class	3
isoclinism	isoclinic class of #253459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2615	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160671);