#251949

order	512
nilpotency class	3
isoclinism	isoclinic class of #251935
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	#3006	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

847 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)(9,13)(10,12)(11,14)(15,16)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,42)(28,44)(29,39)(31,47)(32,41)(35,40)(37,43)(38,46)(45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,22,29)(18,23,26,39)(20,31)(21,28,36,46)(24,37)(25,38,34,44)(27,32)(30,42)(33,45)(35,43)(40,48)(41,47);(6,12)(13,16)(17,20)(18,24)(19,27)(21,35)(22,32)(23,37)(25,41)(26,40)(28,45)(29,31)(30,34)(33,36)(38,47)(39,48)(42,44)(43,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(21,36)(25,34)(28,46)(30,41)(33,35)(38,44)(42,47)(43,45)

Polycyclic presentation:

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

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, 159145);