#121656

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

458 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28852);