#120939

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

403 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,10)(4,12,14,11)(5,8)(6,9)(7,13,15,16)(17,18,21,26,22,25,36,42)(19,27,30,43,29,23,35,38)(20,31,34,45,33,24,39,41)(28,37,44,47,32,40,46,48);(1,3,5,2,6,8,10,9)(4,12,13,15,14,11,16,7)(17,19)(18,23)(20,32,33,28)(21,35)(22,29)(24,40,31,37)(25,27)(26,43)(30,36)(34,44,39,46)(38,42)(41,47,45,48);(1,4,5,13,6,14,10,16)(2,7,3,11,9,15,8,12)(17,20)(18,24,25,31)(19,28)(21,34)(22,33)(23,37,27,40)(26,41,42,45)(29,32)(30,44)(35,46)(36,39)(38,47,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28135);