#268280

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

822 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 175476);