#97367

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

446 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi

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, 4563);