#93603

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

196 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 82 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:

(1,2,4,6,5,8,3,7)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22)(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);(2,6)(3,4)(7,8)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21)(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=fhi,d^2=g,e^2=fi,f^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,c^b=cg,d^b=dh

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, 799);