#149893

order	512
nilpotency class	5
isoclinism	isoclinic class of #149893
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11542
prime radical	#585	C₂×C₄
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57089);