#108407

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gh,c^b=ci,f^b=fh,f^c=fh

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, 15603);