#108323

order	512
nilpotency class	3
isoclinism	isoclinic class of #108316
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

766 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

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

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, 15519);