#298384

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	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 915 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)(23,32)(24,36)(25,30)(26,40)(27,31)(28,34)(29,39)(33,50)(35,44)(37,43)(38,48)(41,45)(42,47)(46,51)(49,52);(1,2,4,3)(5,7,10,16)(6,11,13,20)(8,18,12,14)(9,19,17,15)(21,23)(22,28)(24,37)(25,34)(26,35)(27,41)(29,46)(30,32)(31,44)(33,39)(36,49)(38,52)(40,45)(42,51)(43,48)(47,50);(5,8,9,6,10,12,17,13)(7,14,15,11,16,18,19,20)(21,24,26,39,25,38,27,42)(22,29,31,36,30,47,40,48)(23,33,35,52,34,51,41,37)(28,43,44,46,32,49,45,50);(1,3,4,2)(22,30)(24,38)(28,32)(31,40)(33,51)(37,52)(39,42)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=ehi,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, 205580);