#93309

order	512
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 2 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fi,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,d^b=dg

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, 505);