#149894

order	512
nilpotency class	5
isoclinism	isoclinic class of #149894
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

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

normal subgroups only

This group has 246 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=ci,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

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57090);