#98665

order	512
nilpotency class	3
isoclinism	isoclinic class of #98566
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 546 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) = 512

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

(2,5,6,7)(10,11)(12,13)(15,16)(17,18,23,26,21,20,22,25)(19,24,30,32,28,27,29,31);(1,2)(3,7)(4,6)(5,8)(9,10,14,16)(11,12,15,13)(17,19,21,28)(18,24,20,27)(22,30,23,29)(25,32,26,31);(9,11)(10,12)(13,16)(14,15)(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=gi,d^2=h,e^2=g,f^2=gi,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, 5861);