#99289

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
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	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

232 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,6,3)(4,5,8,7)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,35,38,31)(28,39,42,54)(29,34)(30,33)(32,41,47,51)(36,50)(37,49,53,45)(40,55)(43,48)(44,52)(46,56);(1,3,5,8,6,2,7,4)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,37,30,38,43,53)(26,31,34,45,33,35,48,49)(28,40,41,56,42,36,51,52)(32,46,39,50,47,44,54,55);(1,4,5,3,6,8,7,2)(11,17)(14,20)(18,23)(21,24)(25,28)(26,32,33,47)(27,36)(29,41)(30,42)(31,44,35,46)(34,39,48,54)(37,52)(38,40)(43,51)(45,55,49,50)(53,56)

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=cgi,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, 6485);