#348668

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	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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)(17,18,24,28)(19,23,27,22)(20,25,33,41)(21,34,36,30)(26,38,42,35)(29,37,40,32)(31,43,46,44)(39,45,48,47);(2,6)(7,15)(8,13)(9,16)(17,19,22,18,24,27,23,28)(20,29,32,25,33,40,37,41)(21,34,35,42,36,30,38,26)(31,44,45,48,46,43,47,39);(1,3)(2,7)(4,10)(5,14)(6,15)(8,16)(9,13)(11,12)(17,20)(18,25)(19,29)(21,31)(22,32)(23,37)(24,33)(26,39)(27,40)(28,41)(30,43)(34,44)(35,45)(36,46)(38,47)(42,48);(5,12)(8,13)(9,16)(11,14)(17,21)(18,26)(19,30)(20,31)(22,35)(23,38)(24,36)(25,39)(27,34)(28,42)(29,43)(32,45)(33,46)(37,47)(40,44)(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,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, 255864);