#99279

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

276 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,3)(4,5)(7,8)(9,10,11,13,12,14,15,16)(17,18)(19,27,31,23)(20,32,34,47)(21,26)(22,25)(24,33,38,42)(28,35)(29,43)(30,40,41,37)(36,44)(39,48)(45,46);(1,2,4,3,6,8,5,7)(17,19,28,41,22,31,21,30)(18,23,35,40,25,27,26,37)(20,29,42,48,34,45,33,44)(24,36,47,43,38,39,32,46);(1,3,5,2,6,7,4,8)(17,20,22,34)(18,24)(19,29,31,45)(21,33,28,42)(23,36)(25,38)(26,32)(27,39)(30,44,41,48)(35,47)(37,46)(40,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475);