#135733

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1842
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,22,28,23,27,38,44,24,29,37,26,21,39,20,35)(19,25,32,40,33,41,30,45,34,42,46,43,31,47,36,48);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,25)(20,36)(21,31)(22,32)(23,33)(24,34)(26,43)(27,41)(28,40)(29,42)(30,38)(35,48)(37,46)(39,47)(44,45);(2,7)(6,14)(8,12)(9,15)(17,20,21,37,24,38,23,22)(18,26,27,35,29,28,39,44)(19,30,31,32,34,36,33,46)(25,40,41,45,42,43,47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci

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, 42929);