#97028

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1024 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh

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, 4224);