#233817

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1164 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1164 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,11,16)(7,17,8,19)(9,15,10,14)(12,20,13,18)(21,22,27,31)(23,32,36,45)(24,29,39,44)(25,30,26,40)(28,34,42,35)(33,48,51,47)(37,43,38,46)(41,49,52,50);(1,2)(3,4)(5,7)(6,12)(8,11)(9,20)(10,18)(13,16)(14,19)(15,17)(21,23,27,36)(22,28,31,42)(24,33,39,51)(25,35,26,34)(29,41,44,52)(30,45,40,32)(37,50,38,49)(43,47,46,48);(2,4)(5,8,11,7)(6,13,16,12)(9,20,10,18)(14,19,15,17)(21,24,27,39)(22,29,31,44)(23,33,36,51)(25,38,26,37)(28,41,42,52)(30,46,40,43)(32,47,45,48)(34,50,35,49);(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,32,42,45)(29,43,44,46)(33,49,51,50)(41,48,52,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fhi

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, 141013);