#101454

order	512
nilpotency class	3
isoclinism	isoclinic class of #101343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 441 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=gh,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

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, 8650);