#121793

order	512
nilpotency class	3
isoclinism	isoclinic class of #121791
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

474 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 64

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

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

Polycyclic presentation:

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

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, 28989);