#98710

order	512
nilpotency class	3
isoclinism	isoclinic class of #98566
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

722 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5906);