#98854

order	512
nilpotency class	3
isoclinism	isoclinic class of #98848
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

435 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7,8,6)(4,5)(9,10,12,16,13,15,11,14)(18,22)(19,25)(20,28)(24,30)(26,27)(29,31);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,18,20,22,21,24,28,30)(19,23,26,29,27,32,25,31);(1,3)(2,6)(4,8)(5,7)(10,15)(11,12)(17,19)(18,23)(20,26)(21,27)(22,29)(24,32)(25,28)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,e^c=eg,f^c=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6050);