#299323

order	512
nilpotency class	3
isoclinism	isoclinic class of #299301
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×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 174 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:

(1,2,5,7)(3,11,13,16)(4,10,6,9)(8,12,15,14)(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);(2,7)(8,15)(9,10)(11,16)(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);(1,3,4,12,5,13,6,14)(2,8,9,16,7,15,10,11)(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);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,16,15,11)(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=fi,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,c^b=cg,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206519);