#348663

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3531	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255859);