#98081

order	512
nilpotency class	3
isoclinism	isoclinic class of #97250
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	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

559 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,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,e^c=eg,f^c=fgh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5277);