#97051

order	512
nilpotency class	3
isoclinism	isoclinic class of #97048
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

478 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(33,34,38,42,39,36,37,41)(35,43,45,47,46,40,44,48);(1,2,6,9)(3,8,11,16)(4,12,5,7)(10,15,13,14)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,39,46)(34,40,36,43)(37,45,38,44)(41,48,42,47);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(19,25)(22,28)(26,31)(29,32)(33,36,39,34)(35,40,46,43)(37,41,38,42)(44,48,45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4247);