#138504

order	512
nilpotency class	4
isoclinism	isoclinic class of #138488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#29818	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 80 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)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,20,26,21,27,36,41,23,29,37,40,22,28,34,24)(19,30,31,47,38,25,42,43,33,46,35,39,32,44,48,45);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,45)(31,48)(32,38)(35,42)(36,37)(43,46)(44,47);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,39)(26,45)(27,46)(28,30)(29,44)(31,37)(34,42)(36,48)(40,43)(41,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45700);