#97817

order	512
nilpotency class	3
isoclinism	isoclinic class of #97718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 562 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5013);