#149828

order	512
nilpotency class	5
isoclinism	isoclinic class of #149799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11516
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2921 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 224 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,6,12,28,41,7,13,29,42,15,10,27,38)(3,14,17,39,18,8,25,31,19,32,50,61,44,33,49,62)(4,20,23,52,24,36,56,40,22,37,26,57,45,9,34,35)(16,30,46,59,47,51,58,64,21,53,48,60,54,43,55,63);(1,3,15,44,7,19,6,18)(2,8,12,32,13,33,10,14)(4,21,45,47,22,16,24,54)(5,25,28,50,29,49,27,17)(9,30,20,51,36,53,37,43)(11,39,38,62,42,61,41,31)(23,55,56,46,26,58,34,48)(35,63,57,60,40,64,52,59);(1,4,7,22)(2,9)(3,16,19,21)(5,26,29,23)(6,24,15,45)(8,30)(10,37)(11,40)(12,20)(13,36)(14,43)(17,48,50,46)(18,47,44,54)(25,55,49,58)(27,56,28,34)(31,60)(32,51)(33,53)(35,42)(38,52)(39,64)(41,57)(59,62)(61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57024);