#98001

order	512
nilpotency class	3
isoclinism	isoclinic class of #97894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2682	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 565 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5197);