#124847

order	512
nilpotency class	4
isoclinism	isoclinic class of #124838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1507
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7627
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 327 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)(3,13,16,35,17,41,27,54)(4,18,20,9)(5,22,25,10)(6,12)(7,11)(8,15,30,47,31,48,32,23)(14,40,43,61,45,58,44,63)(19,33)(21,50,53,34)(24,36)(26,39,57,37)(28,38)(29,42,59,51,49,46,60,55)(52,62)(56,64);(1,3)(2,8,11,31)(4,14)(5,23)(6,27)(7,17)(9,29,18,49)(10,35,22,54)(12,30,38,32)(13,39,41,37)(15,26)(16,28)(19,51)(20,45)(21,43)(24,55)(25,47)(33,61,36,63)(34,60,50,59)(40,62,58,64)(42,52)(44,53)(46,56)(48,57);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,29)(10,36)(11,18)(12,34)(13,40)(15,46)(16,44)(17,45)(19,25)(22,33)(23,55)(26,56)(27,43)(28,53)(30,60)(31,49)(32,59)(35,63)(37,64)(38,50)(39,62)(41,58)(42,48)(47,51)(52,57)(54,61)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efgi,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, 32043);