#121801

order	512
nilpotency class	3
isoclinism	isoclinic class of #121800
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	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 126 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)(9,10,15,18,14,19,13,28)(11,20,24,31,23,32,22,16)(12,17,26,35,27,36,29,34)(21,33,37,40,38,39,25,30);(1,2,5,3)(4,7,8,6)(9,11,15,24,14,23,13,22)(10,16,28,32,19,31,18,20)(12,25,26,21,27,37,29,38)(17,33,34,30,36,39,35,40);(1,3,5,2)(4,6,8,7)(9,12,15,26,14,27,13,29)(10,17,28,34,19,36,18,35)(11,21,24,37,23,38,22,25)(16,30,32,39,31,40,20,33)

Polycyclic presentation:

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

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, 28997);