#348665

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

Polycyclic presentation:

2^9|a^2=hi,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, 255861);