#100049

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#496	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 485 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7245);