#121716

order	512
nilpotency class	3
isoclinism	isoclinic class of #121715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 488 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,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, 28912);