#299786

order	512
nilpotency class	3
isoclinism	isoclinic class of #299765
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 834 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206982);