#249798

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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 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)(3,12)(4,14)(5,8)(6,11)(7,10)(9,16)(13,15)(17,18,22,26)(19,27,29,41)(20,24,31,39)(21,33,34,25)(23,35,38,30)(28,44,45,43)(32,42,36,40)(37,47,48,46);(1,3,7,13)(2,5,10,4)(6,14,16,8)(9,15,11,12)(20,31)(21,34)(24,39)(25,33)(27,41)(28,45)(30,35)(37,48);(1,4,7,5)(2,8,10,14)(3,9,13,11)(6,15,16,12)(17,19)(18,23)(20,28)(21,35)(22,29)(24,37)(25,41)(26,38)(27,33)(30,34)(31,45)(32,47)(36,46)(39,48)(40,43)(42,44);(1,5,7,4)(2,3,10,13)(6,15,16,12)(8,9,14,11)(17,20,22,31)(18,24,26,39)(19,28,29,45)(21,36,34,32)(23,37,38,48)(25,42,33,40)(27,43,41,44)(30,47,35,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,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, 156994);