#116071

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23267);