#100007

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

439 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7203);