#97311

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2682	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

490 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4507);