#268263

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

1018 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1018 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,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, 175459);