#128781

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
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×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 478 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) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35977);