#124540

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7594
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,16,40,18,41,17,34)(4,19,21,52)(5,23,27,10)(6,12)(7,11)(8,15,30,24,31,47,38,49)(9,22,33,55)(14,42,44,61,45,64,48,39)(20,53)(25,51)(26,36)(28,57,60,37)(29,58,50,46,62,54,63,43)(32,56)(35,59);(1,3)(2,8)(4,14)(5,24)(6,16)(7,18)(9,29)(10,34)(11,31)(12,38)(13,37)(15,28)(17,26)(19,50)(20,54)(21,45)(22,48)(23,40)(25,58)(27,49)(30,36)(32,64)(33,62)(35,42)(39,53)(41,57)(43,56)(44,55)(46,59)(47,60)(51,61)(52,63);(1,4)(2,9,11,33)(3,14)(5,25)(6,22)(7,21)(8,29,31,62)(10,35,23,32)(12,19,36,52)(13,39,41,61)(15,46)(16,48)(17,44)(18,45)(20,27)(24,58)(26,55)(28,59)(30,63,38,50)(34,42,40,64)(37,53,57,51)(43,47)(49,54)(56,60)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 31736);