#108609

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

424 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,c^b=ch,f^b=fhi,f^c=fhi

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, 15805);