#268277

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

999 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi

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, 175473);