#97309

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	#2733	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 492 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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, 4505);