#97014

order	512
nilpotency class	3
isoclinism	isoclinic class of #97012
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	#2819	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

582 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg

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, 4210);