#348089

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	#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; only the 152 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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,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, 255285);