#250049

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18569	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 157245);