#108248

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

815 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 815 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15444);