#98040

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 98 normal subgroups are shown.

normal subgroups only

This group has 547 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=g,f^2=ghi,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, 5236);