#108395

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 530 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)(5,6,9,12,8,11,16,19)(7,13,15,20,14,10,17,18)(21,22)(23,32,35,46)(24,29)(25,30)(26,40)(27,31)(28,34,42,36)(33,45,48,52)(37,43)(38,50)(39,44)(41,47,51,49);(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,17)(19,20)(21,23,26,36,27,35,25,34)(22,28,30,32,31,42,40,46)(24,33,38,49,39,48,37,47)(29,41,43,45,44,51,50,52);(2,4)(7,14)(10,13)(15,17)(18,20)(21,24,25,37,27,39,26,38)(22,29,30,43,31,44,40,50)(23,33,34,47,35,48,36,49)(28,41,32,45,42,51,46,52)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=di,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,g^a=gh,c^b=ci,f^b=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, 15591);