#265315

order	512
nilpotency class	3
isoclinism	isoclinic class of #265313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

935 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)(9,14)(10,13)(11,15)(12,16)(17,18)(19,27)(20,34)(21,38)(22,29)(23,28)(24,30)(25,32)(26,36)(31,47)(33,42)(35,45)(37,46)(39,43)(40,44)(41,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,22,32,24,33,23,21)(18,25,28,27,30,38,29,42)(20,31,36,35,37,48,40,39)(26,41,34,45,44,47,46,43);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,26)(19,31)(21,39)(22,36)(23,40)(24,37)(25,41)(27,45)(28,34)(29,46)(30,44)(32,35)(33,48)(38,47)(42,43);(4,11)(7,15)(9,16)(12,14)(17,21,23,33,24,32,22,19)(18,27,29,25,30,42,28,38)(20,35,40,31,37,39,36,48)(26,43,46,47,44,45,34,41)

Polycyclic presentation:

2^9|b^2=e,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172511);