#150001

order	512
nilpotency class	5
isoclinism	isoclinic class of #149919
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#384	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=efh,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57197);