#135070

order	512
nilpotency class	4
isoclinism	isoclinic class of #135037
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=gi,e^2=hi,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^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42266);