#98002

order	512
nilpotency class	3
isoclinism	isoclinic class of #97894
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

498 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5198);