#119440

order	512
nilpotency class	3
isoclinism	isoclinic class of #119436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 824 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) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ei,g^c=gi

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, 26636);