#97769

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	#2856	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 629 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) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,f^2=gh,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, 4965);