#298367

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 897 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)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22,27,31)(23,28,34,45)(24,35,37,48)(25,39,26,30)(29,41,47,38)(32,46,49,52)(33,44,40,43)(36,51,50,42);(1,2,4,3)(7,14)(8,17)(11,16)(13,19)(21,23)(22,28)(24,36)(25,33)(26,40)(27,34)(29,46)(30,43)(31,45)(32,38)(35,51)(37,50)(39,44)(41,49)(42,48)(47,52);(5,7,8,15,9,14,17,20)(6,10,11,13,12,18,16,19)(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=hi,b^2=e,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, 205563);