#114640

order	512
nilpotency class	3
isoclinism	isoclinic class of #114631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

840 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21836);