#298368

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

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, 205564);