#122519

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	#466	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

265 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=fi,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=dfhi,e^b=ehi,f^b=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29715);