#98032

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

547 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=hi,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, 5228);