#249266

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1168 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 156462);