#149717

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11479
prime radical	#3147	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56913);