#122023

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

672 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29219);