#126692

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,7,10,8,9,11,12)(13,14)(15,26,29,39)(16,22)(17,34)(18,35)(19,36)(20,25)(21,28,38,30)(23,33)(24,32)(27,41,31,42)(37,43,40,44);(2,4)(13,15,18,30,19,27,32,44,20,29,17,28,16,31,33,43)(14,21,24,39,22,40,35,41,25,38,23,26,36,37,34,42);(6,9)(10,12)(13,16,20,19)(14,22,25,36)(15,27,29,31)(17,32,18,33)(21,37,38,40)(23,34,24,35)(26,41,39,42)(28,43,30,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,c^b=ci,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33888);