#103144

order	512
nilpotency class	3
isoclinism	isoclinic class of #103025
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

560 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,4,7)(3,5,8,6)(10,14)(11,12)(15,16)(17,18,21,19)(20,22,24,23);(2,5)(3,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,18)(20,23,24,22);(10,14)(11,12)(15,16)(18,22)(19,23)(20,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10340);