#299526

order	512
nilpotency class	3
isoclinism	isoclinic class of #299505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,5)(4,11)(6,10)(7,13)(8,14)(9,15)(12,16)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,45,33,39)(42,48,43,46);(1,3,5,2)(4,9,12,7)(6,10,14,8)(11,13,16,15)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,29,35,48,36,33,34,46)(26,39,47,43,44,45,32,42);(1,4,6,13,5,12,14,15)(2,7,8,11,3,9,10,16)(17,19)(18,25)(20,33)(21,30)(22,38)(23,31)(24,37)(26,43)(27,41)(28,40)(29,36)(32,39)(34,48)(35,46)(42,44)(45,47);(2,3)(4,12)(8,10)(13,15)(17,20,22,35,23,36,21,34)(18,26,37,47,28,44,27,32)(19,29,38,48,31,33,30,46)(24,39,40,43,41,45,25,42)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,g^b=gh,d^c=dhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206722);