#97997

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	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

515 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,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, 5193);