#135065

order	512
nilpotency class	4
isoclinism	isoclinic class of #135024
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=di,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287682, #72287691, #72287703
maximal normal subgroup of	#72287682, #72287691, #72287703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42261);