#121680

order	512
nilpotency class	3
isoclinism	isoclinic class of #102079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

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, 28876);