#348646

order	512
nilpotency class	3
isoclinism	isoclinic class of #348082
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	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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,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, 255842);