#99282

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×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

254 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478);