#100518

order	512
nilpotency class	3
isoclinism	isoclinic class of #100251
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

381 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=gh,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7714);