#121854

order	512
nilpotency class	3
isoclinism	isoclinic class of #121851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 391 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:

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

Polycyclic presentation:

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

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, 29050);