#113056

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20252);