#98058

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
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	#36741	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

783 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5254);