#135057

order	512
nilpotency class	4
isoclinism	isoclinic class of #134985
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	C₂×#135	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

265 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,10,12,15,13,16,11,7)(17,18,20,26,23,28,22,40)(19,29,32,24,33,43,30,44)(21,35,34,41,39,48,38,27)(25,37,46,36,42,47,45,31);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(17,19)(18,24,28,44)(20,30)(21,36)(22,32)(23,33)(25,41,42,27)(26,29,40,43)(31,39)(34,37)(35,45,48,46)(38,47);(17,19)(18,25)(20,32)(21,37)(22,30)(23,33)(24,41)(26,46)(27,44)(28,42)(29,35)(31,38)(34,36)(39,47)(40,45)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42253);