#149707

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11477
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

261 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56903);