#99535

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#386
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

438 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6731);