#140920

order	512
nilpotency class	4
isoclinism	isoclinic class of #125098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,f^b=fh

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, 48116);