#97815

order	512
nilpotency class	3
isoclinism	isoclinic class of #97718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2617	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 730 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,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, 5011);